Signed Distance Function

Definition

A signed distance function (SDF) returns, for every 3D point, the distance to the closest surface point. The sign says on which side the point is: negative inside, positive outside, zero on the surface. The surface is the zero level set of the function.

Formula

$$f:{\mathbb{R}}^3\to \mathbb{R},\mathcal{S}=\{\mathbf{x}\in {\mathbb{R}}^3\mid f(\mathbf{x})=0\}$$

Sign convention

Lecture 6.1 (Slide 14) uses the opposite convention: positive inside, negative outside. Both exist; state yours.

Trap

The sign only makes sense for closed (watertight) surfaces. For an open surface there is no consistent inside and outside.

DeepSDF (Park et al., CVPR 2019) learns an SDF with a neural network: given a 3D point, it predicts the signed distance to the closest surface. The shape is stored smoothly and compactly in the learned function.

Appears in

• Lecture 1, 3D representations: surface as the zero level set of a continuous function
• Lecture 1, Neural fields: DeepSDF
• Lecture 5, Distance field: normal $\nabla f$, closest point $\mathbf{p}-f\nabla f$
• Lecture 5, Blending: union = min
• Lecture 5, Point cloud → mesh: SDF + marching cubes
• Lecture 6.1, Fields: occupancy vs. UDF vs. SDF
• Lecture 6.1, Exam task: learned SDF, training vs. inference, why labels are needed
• Lecture 11, 3DShape2VecSet and Hunyuan3D: SDF decoded from a latent vector set

Related concepts

• Neural Field
• Marching Cubes
• Occupancy Field
• Mesh
• Implicit and Explicit Representations