1. 1 L p Centroidal Voronoi Tessellation and its Applications Published in Siggraph 2010 報告者 : 丁琨桓

2. 2 Voronoi Tessellation X2 X3 X1 y || y – X 2 || 2 < || y – X 1 || 2 || y – X 2 || 2 < || y – X 3 || 2 Voronoi cell Restricted Voronoi Tessellation

3. 3 Restricted Delaunay Triangulation Dual graph of a Voronoi tessellation is the Delaunay triangulation

4. 4 Centroidal Voronoi Tessellation x1x1 Each Voronoi vertex x i coincides with its Voronoi cell Ω i Voronoi Tessellation x2x2 x3x3 Ω1Ω1 Ω3Ω3 Ω2Ω2 x1x1 x2x2 x3x3 Ω1Ω1 Ω3Ω3 Ω2Ω2

5. 5 Classical Centroidal Voronoi Tessellation Anisotropy Isotropy

6. 6 Classical Centroidal Voronoi Tessellation stableunstable

7. 7 L p Centroidal Voronoi Tessellation Tranditional(L 2 ) CVT Iso-constours for different distance metrics(L 2 ~L ∞ ) Proposed(L p ) CVT

8. 8 L p Centroidal Voronoi Tessellation L p -CVT is defined as the minimizer of the L p -CVT objective function F L p ||.|| p denotes the L p norm  Domain Ω is the surface of input model

9. 9 L p Centroidal Voronoi Tessellation M y 是用來控制 Voronoi vertex x i 調整位置的權 重矩陣  若透過 SVD 分解 Symmetric tensor field G y 來建立 M y ， i.e. G y = M t y M y ，可產生具有 Anisotropy 特性 的 CVT

10. 10 Anisotropic Surface Remeshing Rrestricted L p -CVT for anisotropic surface remeshing

11. 11 Fully Automatic Feature-Sensitive Remeshing Remeshing surfaces with features is a challenging problem. With a specific definition of per-facet normal anisotropy, the L p -CVT objective function naturally recovers the features. Normal anisotropy f

12. 12 Fully Automatic Feature-Sensitive Remeshing The normal anisotropy M f associated with facet f : N f : Unit normal of facet f s : Importance of normal anisotropy ( s = 5 in this paper)

13. 13 使用 Normal anisotropy 的影響力  讓 Voronoi vertex X 調整後的新位置 X’ 盡可能接近模型表面的切平面  藉此讓鄰近尖銳特徵的 Voronoi vertex 調整到尖銳特徵的位置上 NfNf X X’ Fully Automatic Feature-Sensitive Remeshing 尖銳特徵表面的切平面 與其法向量方向

14. 14 Fully Automatic Feature-Sensitive Remeshing Standard CVTL 2 -CVT with normal anisotropy

15. 15 Fully Automatic Feature-Sensitive Remeshing Remeshing surfaces with self-intersections

16. 16 Variational Quad-Dominant Surface Remeshing Using a value of p that gives a good approximation of the L ∞ norm ( p = 8 ) Algorithm (1) distribute vertices randomly then optimize F L8 (2) for each refinement iteration (3) insert a new vertex at the center of each edge of the Restricted Delaunay Triangulation (4) optimize F L8 (5) compute the Restricted Delaunay Triangulation (6) merge triangles in priority order

17. 17 Variational Quad-Dominant Surface Remeshing L p -CVT (before Restricted Delaunay Triangulation) L p -CVT (before triangle merging)

18. 18 Variational Quad-Dominant Surface Remeshing Bommes et al.2009Ray et al.2006L p -CVT

19. 19 Variational Hex-Dominant Meshing L p -CVT for hex-dominant meshing

20. 20 Variational Hex-Dominant Meshing Variational Hex-Dominant Meshing and comparison with [Mar´echal 2009]

21. 21 Conclusion gap