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L p Centroidal Voronoi Tessellation and its Applications

L p Centroidal Voronoi Tessellation and its Applications. Published in Siggraph 2010 報告者 : 丁琨桓. X3. X1. X2. Voronoi Tessellation. || y – X 2 || 2 < || y – X 1 || 2 || y – X 2 || 2 < || y – X 3 || 2. y. Voronoi cell Restricted Voronoi Tessellation.

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L p Centroidal Voronoi Tessellation and its Applications

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

  2. X3 X1 X2 Voronoi Tessellation || y – X2 ||2 < || y – X1 ||2 || y – X2 ||2 < || y – X3 ||2 y • Voronoi cell • Restricted Voronoi Tessellation

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

  4. Centroidal Voronoi Tessellation Ω1 Ω2 Ω1 Ω2 x1 x1 x2 x2 x3 x3 Ω3 Ω3 Voronoi Tessellation • Centroidal Voronoi Tessellation • Each Voronoi vertex xi coincides with its Voronoi cell Ωi

  5. Classical Centroidal Voronoi Tessellation • Anisotropy Anisotropy Isotropy

  6. Classical Centroidal Voronoi Tessellation stable unstable

  7. LpCentroidal Voronoi Tessellation Tranditional(L2) CVT Proposed(Lp) CVT Iso-constours for different distance metrics(L2~L∞)

  8. LpCentroidal Voronoi Tessellation • Lp-CVT is defined as the minimizer of the Lp-CVT objective function FLp • ||.||p denotes the Lp norm • Domain Ω is the surface of input model

  9. LpCentroidal Voronoi Tessellation • My是用來控制Voronoi vertex xi調整位置的權重矩陣 • 若透過SVD分解Symmetric tensor field Gy來建立My，i.e. Gy = MtyMy，可產生具有Anisotropy特性的CVT

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

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

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

  13. Nf X’ X Fully Automatic Feature-Sensitive Remeshing • 使用Normalanisotropy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 尖銳特徵表面的切平面與其法向量方向

  14. Fully Automatic Feature-Sensitive Remeshing Standard CVT L2-CVT with normal anisotropy

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

  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 FL8 (2) for each refinement iteration (3) insert a new vertex at the center of each edge of the Restricted Delaunay Triangulation (4) optimize FL8 (5) compute the Restricted Delaunay Triangulation (6) merge triangles in priority order

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

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

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

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

  21. Conclusion • gap

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