Approximate Catmull-Clark Patches. Scott Schaefer Charles Loop

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1 Approximate Catmull-Clark Patches Scott Schaefer Charles Loop

2 Approximate Catmull-Clark Patches Scott Schaefer Charles Loop Catmull-Clark Surface ACC-Patches

3 Polygon Models Prevalent in game industry Very fast to render Not smooth (faceted) Complicated LOD management High-resolution models require lots of band-width and computational resources

4 Goal: Fast Smooth Surfaces Eliminate faceting artifacts Animate low-res representation Let GPU worry about LOD Current Future

5 DirectX Pipeline Input Assembler Vertex Buffer Index Buffer Vertex Shader Texture Geometry Hull Shader Rasterizer/ Tessellator Interpolator Texture Stream Output Domain Pixel Shader Texture Output Merger Depth/Stencil Render Target

6 Tessellator Unit Tessellator

7 Tessellator Unit Tessellator

8 Domain Shader Tessellation factor per edge Called for each vertex of the sample pattern Early form in the XBox 360 DS_OUT DS(float2 uv : BARYCENTRIC, int patchind : INDEX) { DS_OUT Out = (DS_OUT)0; // fetch data for patch #patchind // evaluate patch at uv } return Out;

9 Subdivision Surfaces Used in movie and game industries Supported by most 3D modeling software Toy Story Disney / Pixar Geri s Game Pixar Animation Studios

10 Subdivision Surfaces Set of rules S that recursively act on a shape p 0 p k+1 = S p k Arbitrary topology surfaces Smooth everywhere 0 p 1 p k p

11 Catmull-Clark Surfaces 6 ) 3( u u u 6 3 u u u 6 ) (1 3 u 6 ) (1 3 v v v 6 ) 3( v v v 6 3 v

12 Catmull-Clark Surfaces

13 Catmull-Clark Surfaces

14 Catmull-Clark Surfaces

15 Catmull-Clark Surfaces

16 Catmull-Clark Surfaces

17 Catmull-Clark Surfaces

18 Catmull-Clark Surfaces

19 Stam s Exact Evaluation Algorithm Subdivide until x is in ordinary region S i P x

20 Stam s Exact Evaluation Algorithm Subdivide until x is in ordinary region S i P x

21 Stam s Exact Evaluation Algorithm Subdivide until x is in ordinary region S i P x

22 Stam s Exact Evaluation Algorithm Subdivide until x is in ordinary region S i P Extract B-spline control points and evaluate at x x

23 Stam s Exact Evaluation Algorithm Subdivide until x is in ordinary region V i V 1 P Extract B-spline control points and evaluate at x x

24 Performance Issues Limits # extraordinary verts

25 Performance Issues Limits # extraordinary verts

26 Performance Issues Limits # extraordinary verts Lots of shader constants V i ( V 1 P ) Valence Constants

27 Solution: Polynomial Patches Replace extraordinary patches with polynomials Geometry patch (degree 3x3) 2 Tangent patches (degree 3x2) Based on conversion from B-spline to Bezier form

28 Solution: Polynomial Patches Replace extraordinary patches with polynomials Geometry patch (degree 3x3) 2 Tangent patches (degree 3x2) Based on conversion from B-spline to Bezier form b03 b13 b23 b33 b02 b12 b22 b32 b01 b11 b21 b31 b00 b10 b20 b30

29 Solution: Polynomial Patches Replace extraordinary patches with polynomials Geometry patch (degree 3x3) 2 Tangent patches (degree 3x2) Based on conversion from B-spline to Bezier form u03 u u u02 u u u01 u11 u21 u00 u u 10 20

30 Solution: Polynomial Patches Replace extraordinary patches with polynomials Geometry patch (degree 3x3) 2 Tangent patches (degree 3x2) Based on conversion from B-spline to Bezier form v02 v12 v22 v32 v01 v11 v21 v31 v00 v10 v20 v30

31 ACC Geometry Patches Use knot-insertion rules from ordinary case At corners, use limit masks

32 ACC Geometry Patches Use knot-insertion rules from ordinary case At corners, use limit masks

33 ACC Geometry Patches Use knot-insertion rules from ordinary case At corners, use n limit masks n 5 2 n n 5 n 5

34 ACC Geometry Patches Use knot-insertion rules from ordinary case At corners, use n limit masks n n 5 2( n 5) 2 n n 5 2( n 5)

35 ACC Geometry Patches Use knot-insertion rules from ordinary case At corners, use n( n 5) limit masks n 4 n n( n 5) n( n 5) n( n 5) n( n 5) 4 n( n 5) n( n 5) 1 n( n 5) 4 n( n 5) 1 n( n 5) 4 1 n ( n 5) n( n 5)

36 ACC Geometry Patches Use knot-insertion rules from ordinary case At corners, use limit masks Smooth everywhere except edges touching extraordinary vertices 1 n( n 5) 4 n( n 5) 1 n( n 5) 4 n( n 5) 1 n( n 5) 1 n( n 5) 4 n( n 5) 4 n( n 5) n n 5 1 n( n 5) 4 n( n 5) 4 1 n ( n 5) n( n 5)

37 ACC Tangent Patches Continuous normal field needed for displacement mapping Create two tangent patches (u/v) for each position patch

38 ACC Tangent Patches Continuous normal field needed for displacement mapping Create two tangent patches (u/v) for each position patch

39 ACC Tangent Patches Continuous normal field needed for displacement mapping Create two tangent patches (u/v) for each position patch v

40 ACC Tangent Patches Continuous normal field needed for displacement mapping Create two tangent patches (u/v) for each position patch v

41 ACC Tangent Patches Continuous normal field needed for displacement mapping Create two tangent patches (u/v) for each position patch v

42 ACC Tangent Patches v10 v00 v 20 v 30 u u u 20 ˆv 10 ˆv 20 ˆv 00 ˆv 30 v( t) u( t) vˆ( t) u( t) v

43 ACC Tangent Patches v10 v00 v 20 v 30 u u u 20 v v ˆv 10 ˆv 20 ˆv ci c c 0 0 u u cos 2 n i c u 1 c u 1 ˆv v

44 Examples

45 Examples ACC Geometry Patches

46 Examples ACC Geo/Tan Patches

47 Examples Catmull-Clark Surface

48 Examples

49 Examples ACC Geometry Patches

50 Examples ACC Geo/Tan Patches

51 Examples Catmull-Clark Surface

52 Examples

53 Examples ACC Geometry Patches

54 Examples ACC Geo/Tan Patches

55 Examples Catmull-Clark Surface

56 Examples ACC Geo/Tan Catmull-Clark

57 Examples

58 Examples ACC Geometry Patches

59 Examples ACC Geo/Tan Patches

60 Examples Catmull-Clark Surface

61 Comparison Better Worse

62 Comparison Better Worse

63 Comparison Better Worse

64 Comparison Better Worse

65 Conclusions Creates visually smooth surfaces Suitable for displacement/normal mapping Handles any number of extraordinary vertices Simple to evaluate Demo part of DirectX March 2008 SDK

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