Neural SDFs for 3D Surfaces

Neural SDFs train a small network to output signed distance, which gives clean watertight surfaces from photos.

Why Does This Exist?

Density NeRFs render great views but extract noisy meshes because density has no surface constraint. Signed distance functions define surfaces exactly at zero with unit gradients. Neural SDFs (NeuS, VolSDF, 2021) predict $f(p)$ with an MLP, convert distance into rendering weights, and supervise from photos. The result meshes cleanly.

Think of It Like This

Fog sculpted by a distance rule into marble

Density NeRF is fog you photograph. Neural SDF is the same fog with a rule: fog must thicken exactly at zero distance and thin by the ruler elsewhere. Carving the zero shell leaves marble instead of cotton.

Eikonal loss is the ruler and rendering weights are the carving. The analogy stops at supervision: photos guide the carving, not hands.

How It Actually Works

MLP $f(p)$ outputs signed distance. VolSDF turns it into density $\sigma(p) = \alpha \cdot \text{sigmoid}(-\beta f(p))$ with sharpness $\beta \approx 50$ . Rendering uses the same volume sum as NeRF. Loss adds Eikonal $\mathbb{E}[(\lVert \nabla f \rVert - 1)^2]$ plus photometric error. Implicit surfaces extraction runs marching cubes at $f = 0$ .

Worked example

For unit sphere truth at $p = (1.1, 0, 0)$ , target $f = 0.1$ . Network predicts $0.12$ with gradient $(1.05, 0.02, 0)$ , norm $1.05$ . Eikonal penalty is $(1.05-1)^2 = 0.0025$ . Density is $\sigma = 50 \times \text{sigmoid}(-50 \times 0.12) \approx 50 \times 0.0025 = 0.125$ , nearly free space, while at $p = (0.99, 0, 0)$ the density spikes past $20$ . That sharp step is the surface.

Watch Out For

Eikonal weight washing out thin structures

A heavy Eikonal term straightens a $0.02$ m wire into a blob. The symptom is missing rails and cords. Lower the weight 10-fold for thin scenes and verify wires before walls.

The Quick Version

Neural SDFs predict distance, not raw density, for exact surfaces.
Eikonal loss keeps gradients near unit length everywhere.
Density for rendering derives from distance through a sharp transform.
Do not use them for smoke or glass; density fields fit volumes better.
Marching cubes at zero extracts the watertight mesh directly.