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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

LpCentroidal Voronoi Tessellation and its Applications

Published in Siggraph 2010

報告者 : 丁琨桓

voronoi tessellation

X3

X1

X2

Voronoi Tessellation

|| y – X2 ||2 < || y – X1 ||2

|| y – X2 ||2 < || y – X3 ||2

y

  • Voronoi cell
  • Restricted Voronoi Tessellation
restricted delaunay triangulation
Restricted Delaunay Triangulation

Dual graph of a Voronoi tessellation is the Delaunay triangulation

centroidal voronoi tessellation
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
classical centroidal voronoi tessellation
Classical Centroidal Voronoi Tessellation
  • Anisotropy

Anisotropy

Isotropy

l p centroidal voronoi tessellation
LpCentroidal Voronoi Tessellation

Tranditional(L2) CVT

Proposed(Lp) CVT

Iso-constours for different distance metrics(L2~L∞)

l p centroidal voronoi tessellation1
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
l p centroidal voronoi tessellation2
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
anisotropic surface remeshing
Anisotropic Surface Remeshing

Rrestricted Lp-CVT for anisotropic surface remeshing

fully automatic feature sensitive remeshing
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

fully automatic feature sensitive remeshing1
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)
fully automatic feature sensitive remeshing2

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

fully automatic feature sensitive remeshing3
Fully Automatic Feature-Sensitive Remeshing

Standard CVT

L2-CVT with normal anisotropy

fully automatic feature sensitive remeshing4
Fully Automatic Feature-Sensitive Remeshing

Remeshing surfaces with self-intersections

variational quad dominant surface remeshing
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

variational quad dominant surface remeshing1
Variational Quad-Dominant Surface Remeshing

Lp-CVT

(before Restricted Delaunay Triangulation)

Lp-CVT(before triangle merging)

variational quad dominant surface remeshing2
Variational Quad-Dominant Surface Remeshing

Lp-CVT

Ray et al.2006

Bommes et al.2009

variational hex dominant meshing
Variational Hex-Dominant Meshing

Lp-CVT for hex-dominant meshing

variational hex dominant meshing1
Variational Hex-Dominant Meshing

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