ITS: Implicit Thin Shell for Polygonal Meshes

TL;DR

The Implicit Thin Shell~(ITS), a concept designed to implicitly represent the sandwich-walled space surrounding~$\mathcal{M}$, is introduced and a tri-variate tensor-product B-spline is employed to represent~$f, coupled with adaptive knot grids that adapt to the inherent shape variations of~$\mathcal{M}$, while restricting~$f$'s basis functions to the first degree.

Abstract

In computer graphics, simplifying a polygonal mesh surface~$\mathcal{M}$ into a geometric proxy that maintains close conformity to~$\mathcal{M}$ is crucial, as it can significantly reduce computational demands in various applications. In this paper, we introduce the Implicit Thin Shell~(ITS), a concept designed to implicitly represent the sandwich-walled space surrounding~$\mathcal{M}$, defined as~$\{\textbf{x}\in\mathbb{R}^3|ε_1\leq f(\textbf{x}) \leq ε_2, ε_1< 0, ε_2>0\}$. Here, $f$ is an approximation of the signed distance function~(SDF) of~$\mathcal{M}$, and we aim to minimize the thickness~$ε_2-ε_1$. To achieve a balance between mathematical simplicity and expressive capability in~$f$, we employ a tri-variate tensor-product B-spline to represent~$f$. This representation is coupled with adaptive knot grids that adapt to the inherent shape variations of~$\mathcal{M}$, while restricting~$f$'s basis functions to the first degree. In this manner, the analytical form of~$f$ can be rapidly determined by solving a sparse linear system. Moreover, the process of identifying the extreme values of~$f$ among the infinitely many points on~$\mathcal{M}$ can be simplified to seeking extremes among a finite set of candidate points. By exhausting the candidate points, we find the extreme values~$ε_1<0$ and $ε_2>0$ that minimize the thickness. The constructed ITS is guaranteed to wrap~$\mathcal{M}$ rigorously, without any intersections between the bounding surfaces and~$\mathcal{M}$. ITS offers numerous potential applications thanks to its rigorousness, tightness, expressiveness, and computational efficiency. We demonstrate the efficacy of ITS in rapid inside-outside tests and in mesh simplification through the control of global error.

Figures (16)

• Figure 1: Construction of sparse voxel octree (SVO) in a bottom-up manner for a 2D case. Initially, an input polygonal mesh (Figure a) needs to be voxelized into a set of grids with width $2^{-K}$ (Figure b). Subsequently, the minimum number of grids with width $2^{-K}$ must be added to the set (Figure c), ensuring that every four adjacent grids can form a larger grid with width $2^{1-K}$ (the red grids in Figure d). This process continues from Depth-2 to Depth-K (from Figures e to h) with grids being constructed in the same manner, resulting in the hierarchical structure of the SVO (Figure i).
• Figure 2: Top: First-degree univariate B-spline basis functions at different depths (visualized in different colors). Bottom: An overlay view of the basis functions at different depths.
• Figure 3: The approximate SDF of the polygonal mesh using B-Spline functions. The 0-level set curve colored in black closely aligns with the model along the cutting plane boundary, achieving a high degree of alignment with concordance.
• Figure 4: ITS is particularly effective for models with abundant levels of detail. The inner and outer bounding surfaces of the thin shell showcase details and features that closely resemble those of the input model. In Figure (d), it is evident that the model is enveloped with precision.
• Figure 5: Consider an input model normalized to fit within a unit box, comprising 32794 faces and 16323 vertices. The height of the SVOs is denoted by $K$. We plot the dependence of thickness on the parameter $K$, as well as the running time (measured in seconds) with respect to $K$.

Theorems & Definitions (2)

• Proposition 3.1
• Theorem 3.1