Recursive Subdivision Surfaces for Geometric Modeling

Transcription

1 Recursive Subdivision Surfaces for Geometric Modeling Weiyin Ma City University of Hong Kong, Dept. of Manufacturing Engineering & Engineering Management Ahmad Nasri American University of Beirut, Dept. of Mathematics & Computer Science Geometrical Modeling and Processing 2000 Hong Kong 0-2 April, 2000

2 Subdivision surfaces for character animation GERI'S GAME, Pixar Animation Studios won the OSCAR for BEST ANIMATED SHORT FILM of 997, Pixar Animation Studios DeRose 998.

3 Subdivision surfaces for character animation GERI'S GAME, Pixar Animation Studios The control mesh for Geri s head, created by digitizing a full-scale model sculpted out of clay. DeRose 998.

4 Subdivision surfaces for character animation TOY STORY 2, Pixar Animation Studios

5 Subdivision surfaces for character animation A BUG'S LIFE, Pixar Animation Studios

6 Applications: Fillet operations- Zheng Fillet Operations From Zheng

7 Applications: Zhang adaptive subdivision Mesh 334 Mesh 9006

8 Generalization of Doo-Sabin The idea consists of repeatedly applying the set of rules R recursively to subsequently generated polyhedra: Dept. of Math. & Computer Science Hong Kong, 2000

9 Generalization of Doo-Sabin If R is properly chosen, at infinity the sequence of polyhedra converges to a smooth surface. At the limit Dept. of Math. & Computer Science Hong Kong, 2000

10 Generalization of Catmull-Clark 3 iterations of Catmull-Clark Subdivision Dept. of Math. & Computer Science Hong Kong, 2000

11 Generalization of Catmull-Clark 3 iterations of Catmull-Clark Subdivision Dept. of Math. & Computer Science Hong Kong, 2000

12 Catmull-Clark Scheme The idea consists of repeatedly applying the set of rules R recursively to subsequently generated polyhedra: Dept. of Math. & Computer Science Hong Kong, 2000

13 Part. Basic Concepts and Existing Schemes

14 Why Subdivision? Arbitrary Topology

15 Irregular Topology

16 Why Subdivision? Arbitrary Topology Scalability Uniform of Representation Numerical Stability Code Simplicity Interrogation

17 Mask What is a mask? m 3 New vertex n i= 0 = n i= 0 m v i m i i m i m 5 m 0 m 2 m

18 Subdivision Curves

19 Chaikin s Curve Cutting off corners

20 Chaikin s Curve

21 Chaikin s Curve

22 Chaikin s Curve

23 Chaikin s Curve Reisenfeld and Forrest showed that the limit curve is the quadratic B-spline

24 Chaikin s Curve: 2 steps

25 Chaikin s Curve Two steps Split: inserting midpoint

26 Chaikin s Curve Two steps Split: inserting midpoint

27 Chaikin s Curve Two steps Averaging

28 Chaikin s Curve Two steps Computing new vertices /4 V i V i+ 3/4

29 Chaikin s Curve Two steps Computing new vertices /4 V i V i+ 3/4

30 Cubic B-spline Curve:

31 Cubic B-spline Curve:

32 Cubic B-spline Curve:

33 Cubic B-spline Curve:

34 Cubic B-spline Curve:

35 Cubic B-spline Curve:

36 Cubic B-spline Curve V i- /2 /2 V i V i+

37 Cubic B-spline Curve: V i- /8 6/8 V i /8 V i+

38 Subdivision Surfaces

39 Chaikin Tensor Product Regular case

40 Chaikin Tensor Product Curve subdivision: Repeated knot insertion. Chaikin: insert at mid interval

41 Chaikin Tensor Product Regular case: Knot insertion at /2

42 Chaikin Tensor Product Regular case: Knot insertion at /2

43 Chaikin Tensor Product Regular case: Knot insertion at /2

44 Chaikin Tensor Product Regular case: Knot insertion at /

45 Chaikin Tensor Product Regular case: Knot insertion at /

46 Chaikin Tensor Product Original face is replaced by a F-face

47 Chaikin Tensor Product Original edge is replaced by an E-face

48 Chaikin Tensor Product Original vertex is replaced by a V-face Dept. of Math. & Computer Science Hong Kong, 2000

49 Catmull-Clark tensor product Inserting knots at mid-interval in one direction Dept. of Math. & Computer Science Hong Kong, 2000

50 Catmull-Clark tensor product Inserting knots at mid-interval in one direction Dept. of Math. & Computer Science Hong Kong, 2000

51 Catmull-Clark tensor product Inserting knots at mid-interval in the other direction Dept. of Math. & Computer Science Hong Kong, 2000

52 Catmull-Clark tensor product A new mesh is generated Dept. of Math. & Computer Science Hong Kong, 2000

53 Catmull-Clark tensor product A face generates a F-vertex Dept. of Math. & Computer Science Hong Kong, 2000

54 Catmull-Clark tensor product A face generates a F-vertex Dept. of Math. & Computer Science Hong Kong, 2000

55 Catmull-Clark tensor product An edge generates a E-vertex Dept. of Math. & Computer Science Hong Kong, 2000

56 Catmull-Clark tensor product An edge generates a E-vertex Dept. of Math. & Computer Science Hong Kong, 2000

57 Catmull-Clark tensor product A vertex generates a V-vertex Dept. of Math. & Computer Science Hong Kong, 2000

58 Catmull-Clark tensor product A vertex generates a V-vertex Dept. of Math. & Computer Science Hong Kong, 2000

59 Arbitrary topology Regular case: insert knots (/2,/2) What to do with arbitrary topology? Dept. of Math. & Computer Science Hong Kong, 2000

60 Subdivision surface definition A Recursive Subdivision Surface S can be defined by S = (P 0, R) P 0 is a polyhedral Network and R is a set of rules. P 0 R Rule Rule2 Rule3 Dept. of Math. & Computer Science Hong Kong, 2000

61 Generalization of Doo-Sabin Start with a cube (8 points) Dept. of Math. & Computer Science Hong Kong, 2000

62 Generalization of Doo-Sabin Cut vertices and edges by plane Dept. of Math. & Computer Science Hong Kong, 2000

63 Generalization of Doo-Sabin Repeat cutting process Dept. of Math. & Computer Science Hong Kong, 2000

64 Generalization of Doo-Sabin One more time Dept. of Math. & Computer Science Hong Kong, 2000

65 Generalization of Doo-Sabin The idea consists of repeatedly applying the set of rules R recursively to subsequently generated polyhedra: Dept. of Math. & Computer Science Hong Kong, 2000

66 Generalization of Doo-Sabin If R is properly chosen, at infinity the sequence of polyhedra converges to a smooth surface. At the limit Dept. of Math. & Computer Science Hong Kong, 2000

67 Generalization of Doo-Sabin Doo-Sabin Quadratic: F-Face Dept. of Math. & Computer Science Hong Kong, 2000

68 Generalization of Doo-Sabin Doo-Sabin Quadratic: F-Face E-Face Dept. of Math. & Computer Science Hong Kong, 2000

69 Generalization of Doo-Sabin Doo-Sabin Quadratic: V-Face F-Face E-Face Dept. of Math. & Computer Science Hong Kong, 2000

70 Generalization of Doo-Sabin The new vertices are computed: w i m = α j= 0 ij v j Where α α ii ij = = n + 5 and 4n 3+ 2cos(2π ( i 4n j) / n) Dept. of Math. & Computer Science Hong Kong, 2000

71 Generalization of Doo-Sabin The masks for generating new vertices are: α 3 α n 3 α 2 α n 2 α α n α 0 α i = 3+ 2cos(2πi 4n / n) α 0 = n + 5 4n Dept. of Math. & Computer Science Hong Kong, 2000

72 Doo-Sabin Scheme Produces C surfaces everywhere. It is Quadratic B-spline in the regular setting. Dept. of Math. & Computer Science Hong Kong, 2000

73 Generalization of Catmull-Clark For a face generate a F-vertex F-vertex Dept. of Math. & Computer Science Hong Kong, 2000

74 Generalization of Catmull-Clark For a face generate a F-vertex F-vertex n n n n i= 0 v i n n n n Dept. of Math. & Computer Science Hong Kong, 2000

75 Generalization of Catmull-Clark For an edge generate a E-vertex: average of its two vertices and the F-vertices of the faces sharing the edge. F-vertex E-vertex f j+ v j+ 4 & $ % n v + i vi+ + fi + fi + i= 0 #! " v j f j Dept. of Math. & Computer Science Hong Kong, 2000

76 Generalization of Catmull-Clark For an edge generate a E-vertex: average of its two vertices and the F-vertices of the faces sharing the edge. F-vertex E-vertex Dept. of Math. & Computer Science Hong Kong, 2000

77 Generalization of Catmull-Clark For a vertex generate a V-vertex: average of its F- vertices, E-vertices and the vertex itself F-vertex E-vertex n 2 n 2 v j V-vertex n 2 n n n n n v j v 2 i 2 n i= 0 n i= 0 f i Dept. of Math. & Computer Science Hong Kong, 2000

78 n A = 2 4 n B = n n C = A A A A A B B B B C Catmull-Clark: Quad case Dept. of Math. & Computer Science Hong Kong, 2000

79 Generalization of Catmull-Clark 3 iterations of Catmull-Clark Subdivision Dept. of Math. & Computer Science Hong Kong, 2000

80 Generalization of Catmull-Clark 3 iterations of Catmull-Clark Subdivision Dept. of Math. & Computer Science Hong Kong, 2000

81 Catmull-Clark Scheme The idea consists of repeatedly applying the set of rules R recursively to subsequently generated polyhedra: Dept. of Math. & Computer Science Hong Kong, 2000

82 Catmull-Clark Scheme The scheme (most popular) produces C2 everywhere except at the extraordinary point (C only). Ball-Storry, Peters and Reif, Zorin, Prautzsch and many others studied its analysis. Sabin devised a modified version that produced bounded curvature at the extraordinary points. Dept. of Math. & Computer Science Hong Kong, 2000

83 Non-Uniform Scheme Sederberg et al Dept. of Math. & Computer Science Hong Kong, 2000