12.3 Subdivision Surfaces. What is subdivision based representation? Subdivision Surfaces

Transcription

1 2.3 Subdivision Surfaces What is subdivision based representation? Subdivision Surfaces

2 Multi-resolution (Scalability) One piece representation (arbitrary topology) What is so special? Numerical stability Code Simplicit y Covers both polygon form and surface form (Uniformity) 2

3 One piece representation 3

4 Multi-resolution (Scalability) 4

5 Covers both polygon form and surface form (Uniformity of representation) 5

6 Catmull-Clark Quadrilateral So, just what is a subdivision surface? Doo-Sabin Triangular Loop Butterfly 6

7 Basic Concept (Catmull-Clark Scheme): : vertices from mesh M 0,, : vertices to be generated for M e 5 0 v 0 e 4 0 e 0 e 3 0 Around a vertex v of degree 5 e 2 0 7

8 Basic Concept (Catmull-Clark Scheme): Generating new face points Face point: centroid of each face : face point f 5 v 0 f 4 f 3 f f 2 8

9 Basic Concept (Catmull-Clark Scheme): Generating new edge points e v e f i i i i f : edge point f 5 e 5 v 0 f 4 e 4 e f e 2 f 2 e 3 f 3 9

10 Basic Concept (Catmull-Clark Scheme): Generating new vertex points n v v e 2 n n n i f 2 i : vertex point e 5 f 5 e f v 0 f 4 v e 2 f 2 e 4 e 3 f 3 0

11 Basic Concept (Catmull-Clark Scheme): Forming new edges : vertex point e 5 f 5 e f v 0 f 4 v e 2 f 2 e 4 e 3 f 3

12 Repeatedly refining the control meshes, one gets M 0,M,M 2,M 3, limit surface (subdivision surface) M 0 M M 2 M 3 S = M 2

13 NURBS Catmull-Clark Modeling made much easier. Why? No restrictions on the topology of the control points Local refinement is possible NURBS Catmull-Clark 3

14 Example of control meshes of Catmull-Clark subdivision surfaces 4

15 Can model any kind of special features (by modifying the subdivision rules) 5

16 Most importantly, can represent any shape with just one surface (one piece representation ) One Piece Solid Modeling Multi-Piece 6

17 Is One Piece Representation Good? Data Management: Rendering: Machining: Animation: Simpler More efficient More precise Crack free 7

18 Does this mean the solid modeling area is no longer needed? 8

19 What is subdivision based representation? Subdivision Surfaces? CAD/CAM 9

20 What is missing?. No parameterization 2. No error control 3. No adaptive tessellation 20

21 Without error control No CAD/CAM applications Without parameterization Difficult to perform picking, rendering, texture mapping Without adaptive tessellation Too expensive to use 2

22 A major breakthrough occurred in 998 Jos Stam Parameterization of Catmull-Clark Subdivision Surfaces

23 Work on Subdivision Surface Parameterization. J. Stam (998) 2. D. Zorin, D. Kristjansson (2002) 3. S. Lai, F. Cheng (2005) 23

24 (Discrete Fourier Transform) J. Stam Lai/Cheng Parameterization The Extended Subdivision Diagram 24

25 Applications of the new parameterization technique Surface Evaluation Texture Mapping Boolean Operations Surface Trimming Adaptive Tessellation Animation 25

26 Surface Evaluation Fast, Exact Rendering 26

27 Texture Mapping : Lai and Cheng,

28 Texture Mapping : Lai and Cheng,

29 Texture Mapping : Lai and Cheng,

30 Boolean Operations 2 2: Lai and Cheng,

31 Surface Trimming 2 2: Lai and Cheng,

32 Adaptive Tessellation 3 3: Lai and Cheng,

33 What is error control? 33

34 Error Control: Given ε > 0, when would M n - S < ε? M 0 M M 2 ε Limit Surface M n S = M Cross-Sectional View 34

35 What metric should we use to assess M n - S for an extra-ordinary patch? 35

36 A solution is finally available F. Cheng, G. Chen, J. Yong Subdivision Depth Computation for Catmull-Clark Subdivision Surfaces

37 This work is also important for adaptive subdivision 5. Control Mesh Limit Surface Uniform Subdivision Adaptive Subdivision 5: J. Yong, F. Cheng, (2004) 37

38 Basic Idea: Use unbalanced subdivision 6 to provide smooth transition between areas with different densities : F. Cheng, J. Jaromczyk et al (989) 38

39 Adaptive subdivision: input: a piecewise surface P and a subdivision level assignment S output: a triangular linear approximaiton P** of P Three phases: Phase : define a label for each vertex of P Phase 2: generate a gradrilateral subdivision mesh P* of P Phase 3: convert P* to a triangular linear approximation P** of P 3/26/204 University of Kentucky 39

40 Phase : /* F { f f is a patch of P } */ for each vertex v of P do L (v ) := max( {} { S(f ) f F, v is a vertex of f } ) 3/26/204 University of Kentucky 40

41 Phase 2:. for each vertex v of P do LABEL(v ) := L(v ); 2. for each patch f of P do Subdivide(f ); 3/26/204 University of Kentucky 4

42 Subdivide(f : quadrilateral surface patch); then then if (LABEL(v ) > 0 for more than one vertex of f ) f, f f balanced_sub( ); for i := to 4 do subdivide( ); else if (LABEL(v ) > 0 for only one vertex of f ) unbalanced_sub( f, f, f2, f3 ); for i := to 3 do subdivide( ); f i f i, f, f, 3/26/204 University of Kentucky 42

43 3/26/204 University of Kentucky 43

44 3/26/204 University of Kentucky 44

45 3/26/204 University of Kentucky 45

46 Example of adaptive subdivision Significant savings 46

47 Subdivision surfaces have already been used in Pixar s Renderman Alias Wavefront s Maya Nichimen s Mirai Newtek s Lightwave 3D 47

48 Question: Is subdivision the representation scheme for future visualization & animation applications? 48

49 The End 49

50 Acknowledgement: Research work presented here is supported by NSF (DMS , DMI ). Some datasets are taken from P. Schroeder, D. Zorin, L. Kobbelt, H. Hoppe. 50