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

2. Authors Stefanie Hahmann Main Posts Research
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 Circulant Matrices
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. Subdivision Surface From PhD thesis of Zhang Jinqiao Doo-Sabin 细分方法
Catmull-Clark 细分方法
Loop 细分方法
Butterfly 细分方法
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)= xn-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)= xn-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