Fitting polynomial surfaces to triangular meshes with Voronoi squared distance minimization

Abstract

This paper introduces Voronoi squared distance minimization (VSDM), an algorithm that fits a surface to an input mesh. VSDM minimizes an objective function that corresponds to a Voronoi-based approximation of the overall squared distance function between the surface and the input mesh (SDM). This objective function is a generalization of the one minimized by centroidal Voronoi tessellation, and can be minimized by a quasi-Newton solver. VSDM naturally adapts the orientation of the mesh elements to best approximate the input, without estimating any differential quantities. Therefore, it can be applied to triangle soups or surfaces with degenerate triangles, topological noise and sharp features. Applications of fitting quad meshes and polynomial surfaces to input triangular meshes are demonstrated.

Appendices

Appendix 1: Convergence to squared distance: error bound

This Appendix proves that if X is an $\varepsilon$-sampling of $\mathcal{S}$ then:

$$ \lim_{\varepsilon \rightarrow 0} \tilde{F}_{{\mathcal{T}} \rightarrow {\mathcal{S}}}({\bf X})=F_{{\mathcal{T}} \rightarrow {\mathcal{S}}}({\bf X}) $$

(7)

Lemma 1 Let y be a point of $\mathcal{T}$ and x _i its nearest point in X (see Fig. 21). Let ${d=\parallel {\bf y} - \Uppi_{\mathcal{S}}({\bf y}) \parallel}$ and $\tilde{d}=\parallel {\bf y} - {\bf x}_i \parallel$. Then for $\varepsilon < 2$ the following bound is sharp:

$$ \tilde{d}^2 - d^2 \leq \varepsilon^2\hbox{lfs}(\Uppi_{{\mathcal{S}}}({\bf y}))(\hbox{lfs}(\Uppi_{{\mathcal{S}}}({\bf y}))+d) $$

Fig. 21

Configuration of the nearest point of ${\Uppi_{\mathcal{S}}({\bf y})}$

Proof

Let $\mathcal{B}_{Vor}$ be the ball centered at y passing through x _i . This ball contains no point of X.

Let $\mathcal{B}_{lfs}$ be the ball tangent to $\mathcal{S}$ at ${\Uppi_{\mathcal{S}}({\bf y})}$ on the opposite side of y and of radius ${\hbox{lfs}(\Uppi_{\mathcal{S}}({\bf y}))}$ and c _lfs its center. By definition of lfs, this ball has no point of $\mathcal{S}$ in its interior and therefore contains no point of X.

Finally, let $\mathcal{B}_{\varepsilon}$ be the ball centered at ${\Uppi_{\mathcal{S}}({\bf y})}$ of radius ${\varepsilon\hbox{lfs}(\Uppi_{\mathcal{S}}({\bf y}))}$. Since X is an $\varepsilon$-sampling this ball has to contain a point of X.

Since ${\Uppi_{\mathcal{S}}({\bf y})}$ is the nearest point of y on ${\mathcal{S}, {\bf y}, \Uppi_{\mathcal{S}}({\bf y})}$ and c _lfs are aligned and the problem is completely symmetric around the line joining them. Figure 21 shows a cut containing this axis.

The bound follows from the fact that $\mathcal{B}_{\varepsilon} \not\subset \mathcal{B}_{Vor} \cup \mathcal{B}_{lfs}$. Let p be a point of $\mathcal{B}_{Vor} \cap \mathcal{B}_{lfs}$. This point exists since $\varepsilon < 2$ and $\mathcal{B}_{\varepsilon}$ is not included in $\mathcal{B}_{Vor}$. Using this point the previous condition can be reformulated as $p \in \mathcal{B}_{\varepsilon}$.

Using triangular identities in ${(\Uppi_{\mathcal{S}}({\bf y}),c_{lfs},{\bf p})}$, we have:

$$ {\parallel} \Uppi_{{\mathcal{S}}}({\bf y}) - {\bf p} {\parallel^2}=2 \hbox{lfs}(\Uppi_{{\mathcal{S}}}({\bf y}))^2(1-\cos{\alpha}) $$

(8)

with α being the ${({\bf p},c_{lfs},\Uppi_{\mathcal{S}}({\bf y}))}$ angle. Using the same identities in (x, c _lfs , p) we obtain:

$$ \begin{aligned} &\tilde{d}^2 =(d+\hbox{lfs}(\Uppi_{{\mathcal{S}}}({\bf y})))^2 +\hbox{lfs}(\Uppi_{{\mathcal{S}}}({\bf y}))^2 \\ &\quad\quad-2(d+\hbox{lfs}(\Uppi_{{\mathcal{S}}}({\bf y})))\hbox{lfs}(\Uppi_{{\mathcal{S}}} ({\bf y}))\cos{\alpha} \\ &\tilde{d}^2 - d^2 =2(d+\hbox{lfs}(\Uppi_{{\mathcal{S}}}({\bf y})))\hbox{lfs} (\Uppi_{{\mathcal{S}}}({\bf y}))(1-\cos{\alpha}) \\ \end{aligned} $$

Using Eq. 8, $(1-\cos{\alpha})$ can be replaced:

$$ \tilde{d}^2 - d^2=(d+\hbox{lfs}(\Uppi_{{\mathcal{S}}}({\bf y})))\frac{\parallel \Uppi_{{\mathcal{S}}}({\bf y}) - {\bf p} \parallel^2}{\hbox{lfs}(\Uppi_{{\mathcal{S}}}({\bf y}))} $$

(9)

Finally, since p is inside $\mathcal{B}_{\varepsilon}$, we have:

$$ {\parallel} \Uppi_{{\mathcal{S}}}({\bf y}) - {\bf p} {\parallel} \leq \varepsilon \hbox{lfs}(\Uppi_{{\mathcal{S}}}({\bf y})) $$

(10)

This finally provides the result:

$$ \tilde{d}^2 - d^2 \leq \varepsilon^2\hbox{lfs}(\Uppi_{{\mathcal{S}}}({\bf y}))(\hbox{lfs}(\Uppi_{{\mathcal{S}}}({\bf y}))+d) $$

(11)

This bound is sharp since it is reached whenever $\mathcal{S}$ is exactly $\mathcal{B}_{lfs}$ and x _i is located at p.

This lemma leads to a global bound:

Proposition 1

If $\mathcal{S}$ is different from a plane and bounded, then:

$$ \tilde{F}_{{\mathcal{T}} \rightarrow {\mathcal{S}}}({\bf X}) - F_{{\mathcal{T}} \rightarrow {\mathcal{S}}}({\bf X}) \leq \varepsilon^2\vert{\mathcal{T}}\vert\sigma_{{\mathcal{S}}}(\sigma_{{\mathcal{S}}} +d_{{\mathcal{H}}}({{\mathcal{T}},{\mathcal{S}}})) $$

where ${\sigma_{\mathcal{S}}=\sup\{lfs({\bf x}),{\bf x} \in \mathcal{S}\}}.$

Proof

Since $\mathcal{S}$ is not a plane and bounded, σ exists. In addition, the definition of the Hausdorff distance gives us ${d \leq d_{\mathcal{H}}({\mathcal{T},\mathcal{S}})}$.

$$ \begin{aligned} e &=\tilde{F}_{{\mathcal{T}} \rightarrow {\mathcal{S}}}({\bf X}) - F_{{\mathcal{T}} \rightarrow {\mathcal{S}}}({\bf X}) \\ &=\int\limits_{\mathcal{T}} \min_i\parallel {\bf y} - {\bf x}_i \parallel^2 {\text{d}}{\bf y} - \int\limits_{\mathcal{T}} \parallel {\bf y} - \Uppi_{{\mathcal{S}}}({\bf y}) \parallel^2 {\text{d}}{\bf y} \\ &=\int\limits_{\mathcal{T}} \min_i\parallel {\bf y}-{\bf x}_i \parallel^2 -\parallel {\bf y}-\Uppi_{{\mathcal{S}}}({\bf y}) \parallel^2 {\text{d}}{\bf y} \\ &\leq \int\limits_{\mathcal{T}} \varepsilon^2\sigma_{{\mathcal{S}}} (\sigma_{{\mathcal{S}}}+d_{{\mathcal{H}}}({{\mathcal{T}},{\mathcal{S}}})) {\text{d}}{\bf y} \\ &\leq \varepsilon^2\vert {\mathcal{T}} \vert \sigma_{{\mathcal{S}}} (\sigma_{{\mathcal{S}}}+d_{{\mathcal{H}}}({{\mathcal{T}},{\mathcal{S}}}))\\ \end{aligned} $$

(12)

Appendix 2: Gradients of the symmetric term ${\nabla \tilde{F}_{\mathcal{S} \rightarrow \mathcal{T}}}$

This appendix gives the gradients of the symmetric term ${\tilde{F}_{\mathcal{S} \rightarrow \mathcal{T}}}$ (Sect. 3.3). They are used by our Quasi-Newton solution mechanism (see Sect. 3.4). This term is computed using the Voronoi diagram of the samples {y _j } ^m _j=1 of $\mathcal{T}$, restricted to the surface $\mathcal{S}$. In this setting, the variables are the vertices {x _i } ⁿ _i=1 of the triangles of $\mathcal{S}$. Recall that the restricted Voronoi diagram computes for each seed y _j a triangulated portion of $\mathcal{S}$ corresponding to the intersection of $\mathcal{S}$ with the Voronoi cell of y _j . One of these triangles T is highlighted in Fig. 22. Each triangle contributes a term in the objective function and in the gradient (see Eq. 6 in Sect. 3.4). Since the vertices of the triangles are not the actual variables, one needs to use the Jacobians of these vertices and combine them with the chain rule in order to get the gradient of ${\tilde{F}_{\mathcal{S} \rightarrow \mathcal{T}}}$ with respect to the variables {x _i } ⁿ _i=1 . To compute the Jacobians, three different configurations (a), (b), and (c) need to be distinguished (see Fig. 22). Configuration (c) (surface vertex) is trivial. The gradients are computed as follows for the two other configurations (a) and (b). For the interested reader, the rest of this appendix details the derivations.

Fig. 22

Computing the gradient of the symmetric term ${\nabla \tilde{F}_{\mathcal{S} \rightarrow \mathcal{T}}}$: configurations of the vertices of $Vor({\bf Y})\vert_\mathcal{S}$

c corresponds to the intersection between the bisector of [y ₁,y ₂] and an edge [x ₁, x ₂] of $\mathcal{S}$.

$$ \begin{array}{l} \frac{{\text{d}}{\bf c}}{{\text{d}}{\bf x}_{{1}}}={\bf e}{\bf w}_{1}^t+(1-u){\bf I}_{3\times3}\\ \frac{{\text{d}}{\bf c}}{{\text{d}}{\bf x}_{{2}}} ={\bf e}{\bf w}_{2}^t+u {\bf I}_{3\times3}\\ \end{array} \quad \hbox{ where:} \left\{ \begin{aligned}{\bf e} &=({\bf x}_{{2}}-{\bf x}_{{1}})\\ {\bf n} &=({\bf y}_{{2}} - {\bf y}_{{1}})\\ k &={\bf n}\cdot{\bf e}\\ h &=\frac{1}{2}{\bf n}\cdot({\bf y}_{{1}}+{\bf y}_{{2}})\\ u &=\frac{1}{k}(h-{\bf n}\cdot{\bf x}_{{1}})\\ {\bf w}_{1} &=\frac{1}{k^2}(h-{\bf n}\cdot{\bf x}_{{2}}){\bf n}\\ {\bf w}_{2} &=-\frac{1}{k^2}(h-{\bf n}\cdot{\bf x}_{{1}}){\bf n} \end{aligned} \right. $$

(13)

c corresponds to the intersection between the three bisectors of [y ₁, y ₂], [y ₂, y ₃], [y ₃, y ₁] and a facet (x ₁, x ₂, x ₃) of $\mathcal{S}$.

$$ \begin{array}{l} \frac{{\text{d}}{\bf c}}{{\text{d}} {\bf x}_{{1}}}={\bf e}{\bf w}_{1}^t\\ \frac{{\text{d}}{\bf c}}{{\text{d}} {\bf x}_{{2}}}={\bf e}{\bf w}_{2}^t\\ \frac{{\text{d}}{\bf c}}{{\text{d}} {\bf x}_{{3}}}={\bf e}{\bf w}_{3}^t \end{array} \quad \hbox{ where: } \left\{ \begin{aligned} {\bf e} &=({\bf y}_{{1}}-{\bf y}_{{2}})\times({\bf y}_{{1}}-{\bf y}_{{3}})\\ {\bf n} &=({\bf x}_{{1}}-{\bf x}_{{2}})\times({\bf x}_{{1}}-{\bf x}_{{3}})\\ k &={\bf n}\cdot{\bf e}\\ {\bf w}_{1} &=(({\bf x}_{{2}}-{\bf x}_{{3}})\times({\bf x}_{{1}}-{\bf c})+{\bf n})/k\\ {\bf w}_{2} &=(({\bf x}_{{3}}-{\bf x}_{{1}})\times({\bf x}_{{1}}-{\bf c}))/k\\ {\bf w}_{3} &=(({\bf x}_{{1}}-{\bf x}_{{2}})\times({\bf x}_{{1}}-{\bf c}))/k\\ \end{aligned} \right . $$

(14)

What follows is an explanation of the derivations. For configurations (a) and (b), the point can always be expressed as the intersection between a plane $\mathcal{P}$ and a line $\mathcal{L}$:

• The plane $\mathcal{P}$ defined from a point p and a normal n

• The line $\mathcal{L}$ defined from a point v and a direction e

The intersection c is defined as:

$$ {\bf c}={\bf v}+u{\bf e} \hbox{ with } u=\frac{({\bf p}-{\bf v}).{\bf n}}{{\bf e}.{\bf n}} $$

(15)

To ease further notations, we define k = e.n. In addition, the Jacobian of c with respect to x _i will be denoted $\frac{d{\bf c}}{d{\bf x}_i}$.

2.1 Derivation for configuration (a)

The plane $\mathcal{P}$ is a Voronoi cell facet. This plane is the bisector of two Voronoi seeds y ₁ and y ₂. Its parameters are therefore:

• n = y ₂ − y ₁

• ${\bf {p}}=\frac{{\bf{y}}_1+{\bf {y}}_2}{2}$

This plane is constant with respect to the variables. The edge separates two vertices of X namely x ₁ and x ₂. Its parameters are:

• e = x ₂ − x ₁

• v = x ₁

Considering a small displacement dX of X, the variables become x _i  + dx _i . The new location of the intersection point becomes c + dc. We have:

$$ {\text{d}}{\bf c}={\text{d}}{\bf x}_1+u {\text{d}}{\bf e}+{\text{d}}u {\bf e}=u {\text{d}}{\bf x}_2+(1-u){\text{d}}{\bf x}_1+{\text{d}}u {\bf e} $$

$$ \begin{aligned} k^2{\text{d}}u &=- {\text{d}}{\bf x}_1.{\bf n} k - ({\bf p} - {\bf v}).{\bf n}({\text{d}}{\bf x}_2-{\text{d}}{\bf x}_1).{\bf n} \\ &={\text{d}}{\bf x}_1.{\bf n}({\bf p}-{\bf x}_2).{\bf n} - {\text{d}}{\bf x}_2.{\bf n}({\bf p}-{\bf x}_1).{\bf n} \\ \end{aligned} $$

The Jacobians of u are as follows:

$$ \frac{{\text{d}}u}{{\text{d}}{\bf x}_1}=\left[\frac{1}{k^2}({\bf p}-{\bf x}_2).{\bf n} \right]{\bf n}^t {, } \quad \frac{{\text{d}}u}{{\text{d}}{\bf x}_2} =-\left[\frac{1}{k^2}({\bf p}-{\bf x}_1).{\bf n} \right]{\bf n}^t $$

Finally, the Jacobians of c are:

$$ \frac{{\text{d}}{\bf c}}{{\text{d}}{\bf x}_1}={\bf e}\frac{{\text{d}}u}{{\text{d}}{\bf x}_1}+(1-u)I_{3\times3}, \quad \frac{{\text{d}}{\bf c}}{{\text{d}}{\bf x}_2}={\bf e}\frac{{\text{d}}u}{{\text{d}}{\bf x}_2}+uI_{3\times3} $$

(16)

2.2 Derivation for configuration (b)

The plane $\mathcal{P}$ is a facet of $\mathcal{S}$ defined by its three vertices x ₁, x ₂ and x ₃. Its parameters are:

• n = (x ₁ − x ₂) × (x ₁ − x ₃)

• p = x ₁.

The edge is the intersection of two bisectors defined by three constant seeds y ₁, y ₂ and y ₃. Its parameters are:

• e = (y ₁ − y ₂) × (y ₁ − y ₃)

• v = c

Note that in this case c can be used to define a point on the line since the line is constant. Therefore:

$$ \begin{aligned} {\text{d}}{\bf c} &={\text{d}}u {\bf e} \\ {\text{d}}{\bf n} &={\text{d}}{\bf x}_1\times({\bf x}_2-{\bf x}_3)+{\text{d}}{\bf x}_2\times({\bf x}_3-{\bf x}_1)+{\text{d}}{\bf x}_3\times({\bf x}_1-{\bf x}_2) \\ \end{aligned} $$

$$ \begin{aligned} &k^2{\text{d}}u =k({\text{d}}{\bf p}.{\bf n}+({\bf p}-{\bf v}).{\text{d}}{\bf n}) - \overbrace{({\bf v}-{\bf p}).{\bf n}}^{ ({\bf v},{\bf p})\in{\mathcal{P}}^2\Rightarrow 0 } {\text{d}}k \\ &k {\text{d}}u =({\bf p}-{\bf v}).{\text{d}}{\bf n}+{\text{d}}{\bf x}_1.{\bf n} \\ &\quad=({\bf x}_1-{\bf v}). \sum_{i=1}^3{\text{d}}{\bf x}_i\times({\bf x}_{(i+1)[3]}-{\bf x}_{(i+2)[3]})+{\text{d}}{\bf x}_1.{\bf n} \\ &\quad=\sum_{i=1}^3 {\text{d}}{\bf x}_i .(({\bf x}_{(i+1)[3]}-{\bf x}_{(i+2)[3]}) \times ({\bf x}_1-{\bf v}))+{\text{d}}{\bf x}_1.{\bf n} \\ \end{aligned} $$

The Jacobians of u are:

$$ \begin{aligned} \frac{{\text{d}}u}{{\text{d}}{\bf x}_1} &=\frac{1}{k} \left[({\bf x}_2-{\bf x}_3)\times({\bf x}_1 - {\bf v})+{\bf n}\right]^t \\ \frac{{\text{d}}u}{{\text{d}}{\bf x}_2} &=\frac{1}{k} \left[({\bf x}_3-{\bf x}_1)\times({\bf x}_1 - {\bf v})\right]^t \\ \frac{{\text{d}}u}{{\text{d}}{\bf x}_3} &=\frac{1}{k} \left[({\bf x}_1-{\bf x}_2)\times({\bf x}_1 - {\bf v})\right]^t \\ \end{aligned} $$

And finally the Jacobians of c are

$$ \frac{{\text{d}}{\bf c}}{{\text{d}}{\bf x}_i}={\bf e} \frac{{\text{d}}u}{{\text{d}}{\bf x}_i} $$