Neural Distance Fields Overview

• Neural Distance Fields are implicit representations that encode unsigned distances to model open, non-watertight 3D geometries.
• The architecture employs an encoder-decoder framework that transforms voxelized point clouds into continuous distance fields for precise surface extraction.
• NDFs enhance reconstruction accuracy on raw, non-manifold shapes and support diverse applications in 3D scanning, AR, robotics, and beyond.

Neural Distance Fields (NDFs) are neural implicit representations that encode geometric structure by regressing the (unsigned) distance from any spatial point to the nearest surface, thereby providing a continuous, high-resolution parametrization of arbitrarily complex 3D shapes. Unlike signed distance fields (SDFs), which require the object to be closed and partition the embedding space into “inside” and “outside,” NDFs predict nonnegative values and effectively represent open, non-watertight, or high-genus surfaces. This enables direct modeling of diverse, real-world geometry—including open surfaces, non-manifold structures, and internal substructures—without artificial closure or the need for explicit surface orientation.

1. Fundamentals and Mathematical Formulation

NDFs define the geometry of a surface as the zero level set of a neural function $f: \mathbb{R}^3 \to \mathbb{R}_0^+$, where, for a query point $p \in \mathbb{R}^3$, $f(p)$ approximates the unsigned distance to the nearest point on a target surface $S$:

$UDF(p, S) = \min_{q \in S} \|p - q\|$

The surface is then implicitly represented as:

$\{p \in \mathbb{R}^3 \mid f(p) = 0\}$

The network is trained to minimize the discrepancy between its output and ground-truth or generated unsigned distance values at a set of sample query locations, with ReLU activation ensuring nonnegativity. This “unsigned” approach generalizes implicit function learning to non-closed shapes, with the field remaining well-defined in the presence of open or topologically complex surfaces (Chibane et al., 2020).

2. Architectural Design and Inference Techniques

The canonical NDF architecture employs an encoder–decoder configuration:

• Encoder: Sparse input point clouds are first voxelized. A 3D convolutional neural network processes this grid, extracting multi-scale feature volumes $F_1,\ldots,F_n$, capturing geometric context at different spatial granularities.
• Decoder: For arbitrary spatial queries $p \in \mathbb{R}^3$, the decoder “samples” the features at $p$ through interpolation, producing a latent feature vector that is input to a small multilayer perceptron $\varphi(\cdot)$ which outputs the predicted unsigned distance. The decoder thus implements:

$f_x(p) = \varphi( F_1(p), \ldots, F_n(p) )$

where $f_x$ is the instance-specific distance field parameterized by the encoded features.

The network remains fully differentiable in $p$ away from cut loci, making gradient-based surface extraction feasible. Specifically, surface projection of a given point $p$ is accomplished with gradient descent:

$$q \leftarrow p - f(p) \left(\frac{\nabla_p f(p)}{\|\nabla_p f(p)\|}\right)$$

This iterative update rapidly converges to a point on the predicted surface, enabling dense point cloud sampling and mesh generation through post-processing (e.g., ball‐pivoting, Poisson surface reconstruction).

Rendering is performed via a modified sphere tracing algorithm. Steps are proportional to $\alpha f(p)$ with empirical damping factors ($\alpha$, $\beta$) to maintain stability even for inexact fields (Chibane et al., 2020).

3. Advantages and Theoretical Implications

NDFs generalize neural implicit representations in two principal ways:

• Support for Open and Complex Topologies: Unlike methods such as occupancy networks and SDFs, which encode only closed or “solid” geometry, NDFs represent both open manifolds (e.g., walls, cloth) and highly non-manifold or internal structures (e.g., the seats inside a car model), eliminating the need for watertightness and circumventing geometric artifacts from enforced closure.
• Broadened Applicability: The unsigned construction allows the function class to span curves, open surfaces, and even general regression manifolds, making NDFs suitable for learning multi-target mappings and non-binary solution spaces. Surfaces need not partition the ambient space into “inside” and “outside,” removing a major source of discretization and post-processing artifacts.

This leads to improved reconstruction accuracy without pre-closure preprocessing, and the potential to approximate general manifolds—including those resulting from sparse or multi-modal input—using a single continuous function (Chibane et al., 2020).

4. Experimental Validation and Performance

The core evaluation is performed on ShapeNet “Cars,” under two distinct preprocessing regimes: closed (watertight) and raw (potentially open or fragmentary) meshes.

Findings:

• Closed ShapeNet Models: NDFs achieve competitive performance with state-of-the-art SDF-based methods in Chamfer distance and visual fidelity while matching the geometric accuracy of leading architectures on closed surfaces.
• Raw ShapeNet Models: On unprocessed data, NDFs outperform prior work by reconstructing internal substructures and open details (e.g., car seats, grilles). SDF-based methods collapse interior geometry or hallucinate closure artifacts, whereas NDFs faithfully reconstruct both shell and complex interior.
• General Manifolds: Demonstrations on garment reconstruction and sparse curve fitting further evidence NDFs’ capacity for general open surface approximation and manifold learning.

The architecture yields high-resolution (dense) point cloud reconstruction—limited primarily by computational resources and chosen sampling resolution. Qualitative and quantitative benchmarks confirm the increased representational power and robustness (Chibane et al., 2020).

5. Surface Extraction, Visualization, and Applications

NDFs provide direct methodologies for surface extraction and visualization:

• Dense Sampling: Points are seeded throughout the volume and snapped onto the predicted surface by iterative projection via the gradient field, yielding arbitrarily dense samplings.
• Mesh Generation: Extracted point clouds are optionally meshed using off-the-shelf algorithms (e.g., ball-pivoting) to recover explicit surfaces for downstream applications.
• Surface Normal Computation: $\nabla_p f(p)$ at $p$ is usable as an unnormalized normal vector, sufficient for shading and surface orientation during rendering.
• Rendering: Modified sphere tracing rapidly finds surface intersections by stepping proportional to the predicted unsigned distance, with convergence guarantees even in the presence of inexact geometry.

Applications demonstrated include shape completion from sparse views, direct surface regression for open and non-manifold solids, and high-fidelity mesh generation. The method is particularly amenable to scenarios where raw sensor data contain open or fragmentary geometry (e.g., real-world 3D scanning, perception in robotic and AR systems).

6. Limitations and Research Directions

Several open challenges and research avenues follow from the NDF framework:

• Direct Mesh Generation: Current surface extraction requires an additional meshing step; integrating mesh extraction into the architecture (e.g., as a differentiable module) is not addressed and remains an open challenge.
• Efficient Sampling for Real-Time Application: While NDF inference is fully continuous, coarse-to-fine or adaptive sampling strategies are necessary for efficient reconstruction in time-constrained settings.
• Broader Regression and Function Learning: The extension of NDF principles to learn zero level-sets of general regression functions (beyond 3D geometry) is highlighted as a promising avenue for multi-target and manifold regression tasks.
• Scaling and Memory: Practical scalability, particularly for extremely large or dynamic scenes, requires exploring network capacity, memory trade-offs, and distributed embedding strategies.
• Surface Topology Recovery: While NDFs offer greater flexibility, careful design is needed when the topology is highly intricate or the distinction between closely packed structures is subtle (to avoid merged or spurious surfaces).

Proposed future work includes integration with direct mesh generation pipelines, exploration of faster sampling and inference, application to generalized regression problems, and deployment in real-time and video-based 3D perception (Chibane et al., 2020).

7. Relations to Other Implicit Geometry Methods

NDFs are distinct from SDF-based and occupancy-based neural representations in several critical ways:

Aspect	SDF-based Methods	Occupancy Networks	Neural Distance Fields (NDF)
Surface closure	Required (closed only)	Required (binary)	Not required (open/incomplete OK)
Output space	$\mathbb{R}$ (signed)	$\{0,1\}$ (occupancy)	$\mathbb{R}_0^+$ (unsigned distance)
Topology support	Closed, manifold	Closed	Open, multi-component, non-manifold
Signed info needed	Yes	Yes	No
Surface extraction	Level-set at 0	Threshold, mesh	Level-set at 0, projection by grad

This approach replaces the strict partitioning of space with a more general distance-based encoding, enabling learning on open, complex, or fragmentary shapes—common in both sensory data and creative applications.

Summary

Neural Distance Fields establish an unsupervised, continuous, and high-resolution implicit representation for complex 3D geometry by regressing unsigned distances from arbitrary points to the surface. Freed from the limitation of requiring closed surfaces and inside/outside dichotomies, NDFs extend neural implicit modeling to open shapes, complex topologies, curves, and general manifolds. Their differentiable architecture facilitates gradient-based surface extraction and efficient rendering. Empirical results demonstrate robustness in reconstructing raw shape data where prior SDF- and occupancy-based methods fail or introduce artifacts. Ongoing research aims to automate mesh extraction, extend the paradigm to generalized manifold regression, improve sampling and inference efficiency, and broaden applications to diverse 3D vision and function approximation tasks.