Sharp boundary offset guarantees marching cubes mesh contains surface
A Sharp Conservative Offset for Marching Cubes
Graphics
Summary
When turning 3D data into a surface mesh, it's important to know how close the mesh is to the true surface. The authors found the exact smallest distance you need to offset the surface so the mesh created by the marching cubes method fully contains the real surface. They prove this works even though parts of the mesh faces might cross the true surface. Their finding helps ensure reliable 3D surface extraction from grid data.
What this means in practice
- •For 3d graphics developers: Guarantee that meshes generated from volumetric data fully enclose the target surfaces without gaps or holes.
- •For medical imaging engineers: Ensure accurate skin or organ boundary reconstruction from scanned volumetric data using marching cubes with provable error bounds.
A theory result. No direct application yet.
Authors
Alec Jacobson
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 $f^{-1}(0)$.