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