Bicubic G 1 interpolation of arbitrary quad meshes using a 4-split

1. Bicubic G1 interpolation of arbitrary quad meshes using a 4-split Geometric Modeling and Processing 2008 S. Hahmann G.P. Bonneau B. Caramiaux CAI Hongjie Mar. 20, 2008

2. Authors Stefanie Hahmann • Main Posts Professor at Institut National Polytechnique de Grenoble (INPG), France Researcher at Laboratorie Jean Kuntzmann (LJK) • Research CAGD Geometry Processing Scientific Visualization

3. Authors Georges-Pierre Bonneau • Main Posts Professor at Université Joseph Fourier Researcher at LJK • Research CAGD Visualization

4. Outline • Applications of surface modeling • Background • Subdivision surface • Global tensor product surface • Locally constructed surface • Circulant Matrices • Vertex Consistency Problem • Surface Construction by Steps

5. Applications of Surface Modeling • Medical imaging • Geological modeling • Scientific visualization • 3D computer graphic animation

6. A peep of HD 3D Animation From Appleseed EX Machina (2007)

7. Doo-Sabin 细分方法 Catmull-Clark 细分方法 Loop 细分方法 Butterfly 细分方法 Subdivision Surface From PhD thesis of Zhang Jinqiao

8. Locally Constructed Surface From S. Hahmann, G.P. Bonneau. Triangular G1 interpolation by 4-splitting domain triangles

9. Circulant Matrices • Definition: A circulant matrix M is of the form • Remark: Circulant matrix is a special case of Toeplitz matrix

10. Circulant Matrices • Property: Let f(x)=a0+a1x +…+ an-1xn-1, then eigenvalues, eigenvectors and determinant of M are • Eigenvalues: • Eigenvectors: • Determinant:

11. Examples of Circulant Matrices • Determine the singularity of Solution: f(x)=0.5+0.5xn-1,

12. Examples of Circulant Matrices • Compute the determinant of • Compute the rank of

13. Vertex Consistency Problem • For C2 surface assembling If G1 continuity at boundary is satisfied, then

14. Vertex Consistency Problem • Twist compatibility for C2 surface then

15. Vertex Consistency Problem • Matrix form It is generally unsolvable when n is even

16. Sketch of the Algorithm • Given a quad mesh • To find 4 interpolated bi-cubic tensor surfaces for each patch with G1 continuity at boundary

17. Preparation: Simplification • Simplification of G1 continuity condition

18. Choice of • Let be constant, depended only on n (the order of vertex v) • Specialize G1 continuity condition at ui=0, then • Non-trivial solution require

19. Choice of • Determine ni is the order of vi

20. Step 1:Determine Boundary Curve • Differentiate G1 continuity equation and specialize at ui=0, then • Matrix form

21. Examples of Circulant Matrices • Determine the singularity of Solution: f(x)=0.5+0.5xn-1,

22. Step 1:Determine Boundary Curve • Differentiate G1 continuity equation and specialize at ui=0, then • Matrix form

23. Step 1:Determine Boundary Curve • Notations • Selection of d1,d2

24. Step 2:Twist Computations • d1,d2 is in the image of T • Determine the twist • Determine

25. Change of G1 Conditions • From • To

26. Step 3: Edge Computations • Determine • Determine Vi(ui) where V0,V1 are two n×n matrices determined by G1 condition

27. Step 3: Edge Computations • Determine

28. Step 4: Face Computations • C1 continuity between inner micro faces • We choose A1,A2,A3,A4 as dof.

29. Results

30. Results

31. Conclusions • Suited to arbitrary topological quad mesh • Preserved G1 continuity at boundary • Given explicit formulas • Low degrees (bi-cubic) • Shape parameters control is available

32. Reference • S. Hahmann, G.P. Bonneau, B. Caramiaux Bicubic G1 interpolation of arbitrary quad meshes using a 4-split • S. Hahmann, G.P. Bonneau Triangular G1 interpolation by 4-splitting domain triangles • Charles Loop A G1 triangular spline surface of arbitrary topological type • S. Mann, C. Loop, M. Lounsbery, et al A survey of parametric scattered data fitting using triangular interpolants

33. Thanks!Q&A