Signed Distance Functions (SDFs)

• Signed Distance Functions (SDFs) are continuous, real-valued functions that return the Euclidean distance with a sign to indicate a point's relation to a surface.
• They leverage properties like Lipschitz continuity and the |∇φ|=1 condition, making them ideal for PDE-based surface evolution and neural implicit methods.
• SDFs find practical application in computer graphics, physical simulations, and generative modeling by supporting efficient surface reconstruction, collision detection, and differentiable rendering.

A signed distance function (SDF) is a real-valued function φ : ℝ³ → ℝ that, for any point x in space, returns the distance from x to the nearest point on a surface S, with a sign indicating whether x is inside (typically negative) or outside (positive) with respect to S. The zero-level set {x | φ(x) = 0} implicitly defines the surface geometry, while the magnitude |φ(x)| encodes Euclidean distance to S. SDFs are foundational in numerous computational domains, including geometric modeling, shape reconstruction, classification, rendering, collision detection, and physics-based simulation, due to their compactness, differentiability, and capacity to implicitly encode complex topologies.

1. Mathematical Foundations of Signed Distance Functions

The signed distance function for a closed surface S is defined as:

φ(x) = – d(x, S), if x ∈ interior of S + d(x, S), if x ∈ exterior of S

where d(x, S) is the minimum Euclidean distance from x to S. The function φ is Lipschitz continuous and satisfies |∇φ| = 1 almost everywhere except at singularities (e.g., medial axes or corners), a property formalized by the Eikonal equation:

|∇φ(x)| = 1 for x ∈ ℝ³ (1)

This property makes SDFs amenable to PDE-based techniques such as level-set methods for surface evolution. SDFs generalize naturally to arbitrary codimensions and can be defined analytically for primitive shapes (e.g., spheres, cubes) or constructed numerically for arbitrary surfaces via grid discretization, fast marching methods, or neural approximation schemes.

2. Classical and Neural Representations

Historically, SDFs have been realized through analytical formulas for simple shapes (e.g., D(x) = |x| – r for a sphere (McMillan et al., 2021)), voxel grids, or mesh-based approximations (e.g., via the minimum distance from query points to mesh surface). Discrete SDFs are susceptible to memory inefficiency and limited spatial resolution. Grid-based methods (e.g., Fast Marching) or reinitialization PDEs offer only approximate level-set alignment and may induce numerical errors in the zero-crossing.

Recent advances employ neural networks (fully connected MLPs) to approximate SDFs in continuous domains, conditioned either on coordinates alone (single shape) or in conjunction with low-dimensional latent codes (for a shape family). The DeepSDF framework (Park et al., 2019) parameterizes φ(z, x) with a decoder network and a learned latent code z per shape; MetaSDF (Sitzmann et al., 2020) replaces the latent vector optimization with a meta-initialized parameter set adapted by a few gradient steps for each new shape. Implicit neural SDFs enable sub-voxel accuracy, analytic differentiation for normal computation, and efficient memory sharing across large shape collections.

3. SDFs for Surface Reconstruction from Raw Data

Given noisy or sparse measurements (e.g., unoriented or noisy point clouds), robust SDF reconstruction remains a core challenge. Classical SDF estimation pipelines are limited by reliance on input normals or mesh connectivity. Neural-Pull (Ma et al., 2020) and related approaches define a self-supervised loss by "pulling" query points onto the surface using the predicted SDF value and its gradient, enabling direct fitting from raw point sets. Thin plate spline-based or bilateral filtering methods further regularize the neural implicit field to maintain geometric details at edges and corners (Chen et al., 2023, Li et al., 18 Jul 2024).

Recent methods generalize learning to unsupervised or semi-supervised regimes: GenSDF (Chou et al., 2022) meta-learns shape priors and leverages self-supervised projection losses for handling unlabeled and unseen object classes; noise-to-noise mapping strategies (Ma et al., 2023, Zhou et al., 4 Jul 2024) statistically denoise aggregated raw scans and fit consistent neural SDFs without requiring ground-truth distances or normals, relying on Earth Mover's Distance and geometric consistency regularization. Neural variational methods (Weidemaier et al., 15 Apr 2025) compute SDFs from unoriented point clouds by blending local heat kernel approximations with variational fitting of neural fields, bypassing direct eikonal constraints and ensuring convexity of the optimization landscape.

4. Losses and Regularization in Neural SDF Training

A central technical issue is enforcing the distance property |∇φ| = 1 during learning. Standard practice penalizes deviations via the eikonal loss [e.g., ∫Ω(|∇φ(x)| − 1)²dx], but this is ill-posed and admits infinitely many solutions (e.g., sawtooth functions in 1D). ViscoReg (Krishnan et al., 1 Jul 2025) augments the eikonal loss with a viscosity term εΔφ(x) that regularizes the gradient flow and selects the unique physically meaningful (viscosity) solution, ensuring convergence and stabilizing training. Theoretical work shows that minimizing the regularized residual and boundary loss yields provable convergence to viscosity solutions, with explicit generalization error bounds connecting network training loss and L∞ error in the SDF.

Gradient consistency is crucial for correct level-set geometry—not only should |∇φ| = 1, but the vector field ∇φ should vary smoothly and remain parallel between closely spaced level sets (Ma et al., 2023). Level set alignment losses penalize the angular deviation between the gradients at arbitrary query points and their orthogonal projections onto the zero-level set, thereby reducing surface artifacts and improving reconstruction fidelity in sparse or ambiguous data regimes. Implicit bilateral filtering further exploits local spatial and gradient coherence to recover high-frequency features while smoothing noise (Li et al., 18 Jul 2024).

5. Applications Across Domains

SDFs support a diverse set of applications:

• Computer Graphics & Vision: SDFs enable high-fidelity surface reconstruction for scene capture, provide seamless interpolation in generative modeling (DeepSDF (Park et al., 2019), Diffusion-SDF (Chou et al., 2022)), and underpin differentiable rendering pipelines (Wang et al., 14 May 2024). Real-time SDF generation facilitates soft shadow approximation in interactive environments using hybrid algorithms combining jump flooding and selective ray tracing (Tan et al., 2022).
• Shape Classification and Machine Learning: SDF-based binary classifiers provide robust, geometry-aware decision boundaries that outperform or match SVMs and classical methods in synthetic and real datasets, particularly under label or sampling bias (0812.3147).
• Physical Simulation: For garment and character simulation, SDFs afford fast, analytic collision queries for animated or kinematically skinned avatars. Shallow SDF models (Akar et al., 11 Nov 2024) achieve real-time performance by partitioning the body into regions, each represented by a local, low-capacity neural network and aggregated at query time via a boundary-aware stitching operation.
• Photon Transport and Radiation Modeling: SDF-based representations enable accurate and efficient simulation of light transport in turbid or highly curved geometries for Monte Carlo radiation transfer, surpassing voxel-based methods in both geometric fidelity and propagation speed (McMillan et al., 2021).
• Surface PDEs and Constructive Geometry: The combination of accurate SDFs and analytic gradients enables numerical PDE solvers (e.g., for mean curvature flow (Weidemaier et al., 15 Apr 2025)) on the zero-level set, as well as supporting Boolean operations for constructive solid geometry.

6. Generative and Probabilistic SDF Models

SDFs are now used in probabilistic frameworks for generative modeling. Diffusion-SDF (Chou et al., 2022) employs latent diffusion over variationally encoded SDF representations, supporting both unconditional and conditional 3D shape synthesis, robust completion from partial data, and interpolation in the learned latent manifold. The paradigm of conditioning SDF decoders on latent codes, meta-learned initializations, or gradient-based adaptation underlines a trend toward highly flexible, low-shot inference models (MetaSDF (Sitzmann et al., 2020), GenSDF (Chou et al., 2022)), which not only reduce test-time optimization costs but also generalize across highly varied shape classes.

7. Challenges, Limitations, and Future Directions

Despite their strengths, SDF-based methods face several challenges:

• Ill-posedness and Training Instability: Direct eikonal supervision can lead to unstable or degenerate solutions in neural representations; viscosity regularization and meta-learning-based adaptation offer partial remedies (Krishnan et al., 1 Jul 2025, Sitzmann et al., 2020).
• Noisy/Sparse Data and Unoriented Inputs: Effective SDF inference from real sensor data without access to ground-truth normals or distances remains an area of active research (Chen et al., 2023, Ma et al., 2023, Zhou et al., 4 Jul 2024, Weidemaier et al., 15 Apr 2025).
• Global vs Local Representation: Partitioning complex or articulated domains into locally modeled SDFs and then robustly stitching the results introduces both scalability and accuracy trade-offs (Akar et al., 11 Nov 2024). General-purpose shallow SDF models excel in speed but may require elaborate boundary aggregation for global consistency.
• Diffusion-and Modulation-based Generative Models: While latent diffusion SDF models enable higher-quality 3D generation, advancing beyond object-level to full-scene and appearance modeling is an open frontier (Chou et al., 2022).
• Differentiable Rendering: Visibility-induced nonsmoothness remains challenging for differentiable SDF rendering. Recent progress involves band-relaxation of visibility boundaries, trading controlled bias for much lower variance and efficient integration into optimization pipelines (Wang et al., 14 May 2024).
• Complex Scene Representation and Real-world Robustness: Extending SDF-based techniques to multi-object, dynamic, and multi-modal scenarios, as well as reducing the dependence on canonical pose or fixed input modality, are key directions for future research (Park et al., 2019, Chou et al., 2022).

Table: Representative SDF Methodologies and Characteristics

Method/Class	Key Technical Feature	Application Domain
DeepSDF (Park et al., 2019)	Latent-conditioned neural decoder; auto-decoder; clamped loss	Shape representation, completion
Neural-Pull (Ma et al., 2020)	Self-supervised pulling loss using SDF predictions and gradients	Surface reconstruction
ViscoReg (Krishnan et al., 1 Jul 2025)	Viscous regularization of eikonal loss; stability proofs	Scene reconstruction, theor. analysis
GenSDF (Chou et al., 2022)	Two-stage meta- and semi-supervised learning; zero-shot inference	3D object generalization
Implicit Bilateral Filtering (Li et al., 18 Jul 2024)	Gradient-guided, non-local filtering over levels sets	Detail-preserving surface reconstruction
Diffusion-SDF (Chou et al., 2022)	Conditional/unconditional generative modeling via latent diffusion	Shape synthesis, completion

Each methodology addresses specific challenges—memory efficiency, data bias, gradient consistency, generative diversity—while targeting different operational regimes (offline reconstruction, real-time interaction, generalization to unseen categories, or differentiable optimization).

• (0812.3147) Comparison of Binary Classification Based on Signed Distance Functions with Support Vector Machines
• (Park et al., 2019) DeepSDF: Learning Continuous Signed Distance Functions for Shape Representation
• (Sitzmann et al., 2020) MetaSDF: Meta-learning Signed Distance Functions
• (Ma et al., 2020) Neural-Pull: Learning Signed Distance Functions from Point Clouds by Learning to Pull Space onto Surfaces
• (McMillan et al., 2021) Meshless Monte Carlo Radiation Transfer Method for Curved Geometries using Signed Distance Functions
• (Chou et al., 2022) GenSDF: Two-Stage Learning of Generalizable Signed Distance Functions
• (Chou et al., 2022) Diffusion-SDF: Conditional Generative Modeling of Signed Distance Functions
• (Chen et al., 2023) Unsupervised Inference of Signed Distance Functions from Single Sparse Point Clouds without Learning Priors
• (Ma et al., 2023) Towards Better Gradient Consistency for Neural Signed Distance Functions via Level Set Alignment
• (Ma et al., 2023, Zhou et al., 4 Jul 2024) Noise-to-noise SDF mapping from noisy point clouds
• (Wang et al., 14 May 2024) A Simple Approach to Differentiable Rendering of SDFs
• (Li et al., 18 Jul 2024) Implicit Filtering for Learning Neural Signed Distance Functions from 3D Point Clouds
• (Akar et al., 11 Nov 2024) Shallow Signed Distance Functions for Kinematic Collision Bodies
• (Weidemaier et al., 15 Apr 2025) SDFs from Unoriented Point Clouds using Neural Variational Heat Distances
• (Krishnan et al., 1 Jul 2025) ViscoReg: Neural Signed Distance Functions via Viscosity Solutions