A Sharp Conservative Offset for Marching Cubes

2026-09-11

AI Summary

该研究针对网格采样的1-利普希茨函数，通过确定一个尖锐下界σ*，确保真零水平集被包含在移动立方体网格中，从而解决了准确构建三维模型的问题。

Abstract

Given a grid-sampled 1-Lipschitz function $f:\mathbb{R}^3 \rightarrow \mathbb{R}$ (such as a signed distance function) sampled on a regular grid, we determine the sharp infimum $σ_\star$ such that for every $σ> σ_\star$ the true zero-level set $f^{-1}(0)$ is contained in the marching cubes mesh for $f = σ$. For grids with spacing $h$, the sharp infimum is $σ_\star = \frac{\sqrt{3}}{2} h$. This result is proven \emph{despite} the planar marching cube faces potentially lying on either side of the level set of the trilinearly interpolated approximation of $f$ (which shares the same infimum). The proof instead constructs a convex lift of the corner samples and shows that the high-side portion of each marching-cubes cell lies in the projection of its $σ$-superlevel portion, which cannot contain a point of $f^{-1}(0)$.

Problem

• Marching Cubes
• 1-Lipschitz function
• zero-level set
• grid-sampled

Innovation

• Sharp Conservative Offset
• Marching Cubes
• 1-Lipschitz function
• Trilinear Interpolation