Introduction

Isosurfacing is a method to extract the surface from a three-dimensional scalar field. A scalar field $\varphi =\varphi (x,y,z):{\mathbb{R}}^3\mapsto \mathbb{R}$ assigns a scalar value to every point in three-dimensional space. When every point in the domain sits on a regular (i.e., uniform) three-dimensional grid, the scalar field is a voxel field.

The simplest non-trivial voxel field takes only two integer values, 0 and 1. A 0 marks a point outside the field. A 1 marks a point inside it, or on its boundary. The interface between 0 and 1 everywhere in the grid is the field's isosurface.

Real segmentations are rarely binary. A material segmentation typically assigns a distinct positive integer label to each material, with 0 reserved for void. The isosurface generalizes directly: it is the interface wherever any two labels differ, not only where a label meets void. automesh already works this way — mesh tri produces isosurfaces of the material boundaries, plural, and the cuberille implementation below emits a face wherever two face-adjacent voxels' labels differ, whichever two labels they are. Any isosurfacing method in automesh needs to preserve that: a method that only finds the outer material/void boundary would be a regression, not an upgrade. See Marching Cubes for how Marching Cubes stands on this.

Three approaches, not two

Marching Cubes and Dual Contouring are the two isosurfacing algorithms most often cited in the literature, covered in detail on the next two pages. But a third, older approach exists, and it is the default in automesh.

Cuberille: the mesh tri default

By default, automesh mesh tri (--cubes cuberille) builds its output surface by walking the grid once. For every pair of face-adjacent voxels whose labels differ, the algorithm emits a quad face exactly on the voxel boundary between them — no interpolation, no gradient information. Each quad splits into two triangles, coincident vertices weld together, and any resulting non-manifold "pinch" vertex is duplicated to restore a clean 2-manifold. The result is the exact voxel-boundary surface: axis-aligned, stair-stepped, and topologically faithful to the segmentation it came from.

This is the cuberille method, first described by Herman and Liu¹ in 1979 for reconstructing 3D organ surfaces from CT slices. It predates both Marching Cubes (1987) and Dual Contouring (2002).

Why it matters for mesh tri

Cuberille's stair-stepping is why cuberille output is typically smoothed afterward — mesh tri smooth chains Laplace or Taubin smoothing directly onto it, and mesh tri smooth remesh can follow that with remeshing. The voxel-boundary surface is a faithful but blocky starting point. With cuberille, smoothing, not the isosurfacing step itself, produces a visually smooth result.

Marching Cubes and Dual Contouring take the opposite approach: both interpolate a smoother surface during extraction, at the cost of losing the guarantee that every output vertex sits exactly on a voxel boundary. Marching Cubes interpolates along voxel edges. Dual Contouring places a vertex inside each voxel, using gradient information to better preserve sharp features. Marching Cubes is available through mesh tri --cubes marching. automesh does not implement Dual Contouring.

References

1. Herman GT, Liu HK. Three-dimensional display of human organs from computed tomograms. Computer Graphics and Image Processing. 1979;9(1):1-21. link ↩