Fitting Polynomial Surfaces to Triangular Meshes with Voronoi ...

6 Vincent Nivoliers, Dong-Ming Yan, and Bruno Lévy
F T →S = ∫ T ‖y − Π S(y)‖ 2 dy
≃ ∫ T min i ‖y − x i ‖ 2 dy
= ∑ i
∫Ω i ∩T ‖y − x i‖ 2 dy
where Ω i denotes the 3D Voronoi cell of x i . We shall now give the definition
of the approximation ˜F of F minimized by VSDM :
˜F(X)
where :
˜F T →S
= ˜F T →S (X) + ˜F S→T (X) + λ X t L 2 X
} {{ }
R(X)
= ∑
x i ∈X
˜F S→T = ∑
y j ∈Y
∫
T ∩Ω i
∫
‖ y − x i ‖ 2 dy
S∩Ω j
‖ x − y j ‖ 2 dx
(3)
The matrix L is the uniform graph Laplacian of S. The influence of the regularization
factor λ is illustrated in Figure 3. Ω i denotes the Voronoi cell of
x i in the Voronoi diagram of X, and Ω j the Voronoi cell of y j in the Voronoi
diagram of Y (see Figure 4).
3.2 Convergence to the continuous objective function
The VSDM approximation replaces the nearest point on S with the nearest
sample of X (in the term F T →S ) and the nearest point on T with the nearest
sample of Y (in the term F S→T ). The accuracy of the approximation depends
Fig. 3: Influence of the regularization factor λ on subdivision surface fitting.