Neural Field

Definition

A field maps every point in space to a scalar or vector. A neural field represents it with a neural network that can be queried at any point, e.g. $\Phi :{\mathbb{R}}^3\to \mathbb{R}$ for a signed distance.

Parameterization	Examples	Memory	Speed	Locality	Resolution
Discrete	voxel grid, octree	✗ ($\mathcal{O}(n^d)$)	✓	✓	✗ fixed
Neural	MLP (SIREN, DeepSDF)	✓	✗	✗	✓ any
Hybrid	voxel features, triplane, NGLOD, Instant NGP	trade-off	trade-off	✓	✓

For 3D reconstruction (generalizing across shapes):

• Global code (OccNet, IM-NET, DeepSDF, 2019): $f(\mathbf{z},\mathbf{p})\mapsto [0,1]$. Loses 3D structure and detail.
• IF-Nets (2020): $f({\mathbf{F}}_1(\mathbf{p}),\dots ,{\mathbf{F}}_n(\mathbf{p}))$ with multi-scale local features.
• NDF (2020): unsigned distance output ${\mathbb{R}}^{+}$ for open surfaces; surface by projection $\mathbf{q}=\mathbf{p}-f\nabla f$ or sphere marching.

Intuition

A single neural field is just a network overfit to one scene. NeRF is also a neural field: $(x,y,z,\theta ,\varphi )\mapsto (RGB,\sigma )$.

Appears in

• Lecture 6.1, Neural Fields
• Lecture 6.1, Hybrid Representations
• Lecture 6.1, IF-Nets and NDF
• Lecture 5, Neural distance fields
• Lecture 1, course overview
• Lecture 7, NeRF: a neural radiance field $(x,y,z,\theta ,\varphi )\mapsto (RGB,\sigma )$
• Lecture 11, Structured vs. unstructured latents: voxel/triplane latents vs. vector sets

Related concepts

• Volume Rendering
• Signed Distance Function
• Occupancy Field
• Hash Encoding
• Voxel Grid
• Implicit and Explicit Representations