Marching Cubes

Definition

Marching cubes turns a grid of values of an implicit function (SDF, occupancy, density) into a triangle mesh of the level set $f=0$ (or $f=\tau$).

Algorithm

1. Build a cube from 4 neighbors on one slice and 4 on the next.
2. Index = inside/outside pattern of the 8 corners ($2^8$ cases).
3. Look up the crossed edges in a precalculated table.
4. Place vertices on the edges by linear interpolation of the values.
5. Normals from central differences, interpolated to the triangle vertices.
6. Output the triangles.

Method	Idea	Weakness
Marching cubes (MC)	lookup table + linear interpolation	misses sharp features, cubic cost in resolution
Dual contouring (DC)	one vertex per cell minimizing ${\sum }_i({\mathbf{n}}_i\cdot (\mathbf{x}-{\mathbf{p}}_i){)}^2$ with Hermite data, octree	can give non-manifold vertices, thin features
Dual marching cubes (DMC)	QEF with tangent planes of the field	crack-free, manifold, sharp features
FlexiCubes	extra parameters, differentiable	used to optimize meshes end-to-end

Trap

Classical MC is not differentiable. FlexiCubes is the differentiable variant, not a faster one.

Appears in

• Lecture 5, Marching Cubes
• Lecture 5, Dual Contouring, Dual Marching Cubes, FlexiCubes
• Lecture 5, Point cloud → mesh: the exam pipeline
• Lecture 6.1, Exam task: mesh from a learned SDF
• Lecture 8, SuGaR: naive mesh extraction from 3DGS is noisy

Related concepts

• Signed Distance Function
• Occupancy Field
• Mesh
• Poisson Surface Reconstruction