Signed Distance Function (SDF)

• SDF is a scalar field that encodes the signed Euclidean distance to the nearest surface, with positive values outside, negative inside, and zero on the surface.
• Methodologies include discrete TSDFs, neural implicit representations, and probabilistic models, which are applied in 3D reconstruction, rendering, robotics, and classification tasks.
• Advanced research leverages filtering, regularization, and local code techniques to address numerical instabilities and improve the precision of SDF-based applications.

A Signed Distance Function (SDF) is a scalar field that, for any point in space, encodes the signed Euclidean distance to the nearest surface of an object: positive values indicate points outside the surface, negative values indicate points inside, and zero-valued points lie exactly on the surface. Formally, for a closed region Ω ⊂ ℝ³ with surface ∂Ω, the SDF is

$$\phi(x) = \begin{cases} +\min_{y\in\partial\Omega} \|x - y\|_2, & x \notin \Omega \ -\min_{y\in\partial\Omega} \|x - y\|_2, & x \in \Omega \end{cases}$$

The surface itself is defined as the zero-level set $\{x \mid \phi(x) = 0 \}$. SDFs are foundational in computer graphics, vision, robotics, and computational geometry due to their robust geometrical properties and compatibility with both discrete and neural representations.

1. Mathematical Foundations and Properties

Fundamental properties of true SDFs include 1-Lipschitz continuity ($|\phi(x) - \phi(x')| \leq \|x - x'\|$), differentiability almost everywhere, and the Eikonal equation satisfied almost everywhere:

$|\nabla \phi(x)| = 1$

The gradient $\nabla \phi$ aligns with the surface normal at $\partial\Omega$, and the sign of $\phi$ imparts in/out classification. The SDF admits unique viscosity solutions and, under the Eikonal constraint, provides a mathematically stable definition for signed distance over the whole domain (Krishnan et al., 1 Jul 2025, Fayolle, 2021, Dai et al., 21 Oct 2025).

2. Discrete, Analytical, and Neural Parameterizations

Discrete and Classical Representations

Traditional SDFs are tabulated over voxel grids as in Truncated Signed Distance Functions (TSDFs). Updates fuse multiple sensor measurements by weighted averaging, maintaining a per-cell distance and confidence weight (Millane et al., 2020, Fu et al., 2021). However, discrete storage incurs quantization artifacts, memory overhead, and loss of differentiability.

Neural Implicit SDFs

Recent advances represent SDFs as neural networks: $f_\theta(x) \approx \phi(x)$. The MLPs are trained to regress ground-truth signed distances or to satisfy constraints on input data. Zero-level set extraction is typically performed with marching cubes. Eikonal losses penalize deviations from the unit-norm gradient (Fayolle, 2021, Chou et al., 2022, Li et al., 23 Nov 2024). Modern hybrid methods combine discrete priors (e.g., gradient-augmented octrees) with neural residuals to balance memory efficiency and local fidelity (Dai et al., 21 Oct 2025).

Probabilistic Extensions

Uncertainties are modeled with joint distributions over SDF values and inlier probability, as in Probabilistic SDF (PSDF), incorporating Bayesian fusion and surfel-based meshing (Dong et al., 2018).

3. SDFs in Reconstruction, Rendering, and Generative Modeling

SDFs underpin 3D shape reconstruction from depth, image, or point-cloud data via direct regression, volumetric rendering, or hybrid pipelines. Classical TSDF fusion, geometry-aware loss terms, and neural SDF optimization anchor the field (Millane et al., 2020, Yao et al., 2021). SDF parametrizations have been extended for joint shape and appearance reconstruction through rendering pipelines such as SDF-3DGAN, SplatSDF, and SDF-NeRF, leveraging differentiable rendering and volumetric synthesis for downstream GAN and diffusion modeling (Jiang et al., 2023, Li et al., 23 Nov 2024, Chou et al., 2022).

Differentiable rendering of SDFs enables end-to-end inverse graphics pipelines by relaxing visibility boundaries to thin bands, yielding low-variance, fully differentiable shading gradients suitable for geometry and scene optimization tasks (Wang et al., 14 May 2024). Hybrid approaches combine occupancy fields and SDFs to address bias and representation failures due to multi-surface ambiguity or vanishing gradients in photometrically weak regions (Lyu et al., 2023).

4. Advanced Neural and Geometric Structures

Local and Articulated Codes

DeepSDF and its extensions encode shapes as latent vectors, but global codes are insufficient for highly articulated or complex surfaces. Recent work leverages graph neural networks to distribute local codes across mesh regions for localized SDF approximation and adaptation, improving reconstruction accuracy and expressivity (Yao et al., 2021, Mu et al., 2021).

Articulated SDFs (A-SDF) disentangle global shape and pose into separate latent spaces, enabling smooth control and animation across arbitrary articulation parameters. The decoder consumes both geometric and pose latent codes, yielding SDF predictions and facilitating test-time adaptation for generalization to partial real-world scans (Mu et al., 2021).

Filtering and Regularization

Implicit bilateral filters can be directly embedded in neural SDF optimization, operating on level sets via gradient-based "pulling" to both sharpen and denoise learned fields. This approach outperforms both global smoothness constraints and local averaging, particularly in preserving geometric details such as edges and corners (Li et al., 18 Jul 2024).

Numerically, the Eikonal equation is non-unique and ill-posed for neural optimization. Techniques such as vanishing-viscosity regularization (ViscoReg) introduce Laplacian smoothing for stable, unique viscosity solutions and improved sup-norm generalization error, addressing high-frequency instability in gradient flows (Krishnan et al., 1 Jul 2025).

5. SDF Applications Across Domains

Robotics and Mapping

SDFs serve as the geometric backbone for localization, mapping, and trajectory planning. Accurate and continuous SDF maps—via TSDF, $\nabla$-SDF, or 2D SDF fusion—enable sub-centimeter localization, robust outlier rejection, and memory-efficient scene storage, critical for real-time robot autonomy (Fu et al., 2021, Millane et al., 2020, Dai et al., 21 Oct 2025).

Computer Graphics

In rendering, SDF-based techniques support real-time soft shadow synthesis (RTSDF) through GPU-amenable jump flooding and ray tracing, enabling hybrid, resolution-adaptive representations compatible with dynamic content and interactive frame rates (Tan et al., 2022). SDFs enable efficient GPU-level collision, blending, and constructive solid geometry. Differentiable SDFs further power inverse rendering, relighting, and material estimation (Wang et al., 14 May 2024).

Generative and Probabilistic Models

Recent generative models, such as diffusion-based SDFs and adversarial SDF renderers, exploit neural SDFs as continuous, expressive 3D priors, supporting unconditional synthesis, shape completion, and category-level generative pipelines across partial and multi-modal input signals (Chou et al., 2022, Jiang et al., 2023). Hybrid architecture-level fusions with explicit 3D primitives, e.g., SplatSDF combining SDF MLPs and 3D Gaussian splatting, sharply accelerate convergence and recover photometric/geometric detail (Li et al., 23 Nov 2024).

6. SDFs in Binary Classification and Machine Learning

SDFs can be used as geometric surrogates for indicator functions in nonlinear binary classification. An SDF classifier fits a continuous-valued distance-to-boundary function, enabling decision rules based on the sign of the SDF approximation. The SDF method reformulates classification as kernel regression with geometric target values, in contrast to SVMs' ±1 margin formulation. Empirical results demonstrate that SDF-based classifiers can match or slightly exceed standard SVM and KNN accuracy on both geometric and high-dimensional biomedical tasks, exhibiting improved robustness to class imbalance and computational simplicity due to a unique linear solution (0812.3147).

Approach	SDF Decision Rule	Optimization Type
SDF Classifier	$$\operatorname{sign}(\sum_{i=1}^N \alpha_i K(x,x_i))$$	Linear regression
SVM	$$\operatorname{sign}(\sum_{i=1}^N y_i\alpha_i K(x,x_i) + b)$$	Quadratic program

SDF-based classification leverages geometric information inherent in distance-to-boundary estimation, providing a theoretically grounded and practically robust method that outperforms indicator regression and exhibits advantageous properties in imbalanced-class contexts (0812.3147).

7. Limitations, Open Issues, and Future Directions

While SDFs provide unique advantages—smoothness, differentiability, and implicit topology—their practical deployment can be encumbered by numerical instabilities (e.g., Eikonal ill-posedness), ambiguity in multi-surface scenarios, loss of fine detail at low neural or discrete capacity, or bias in complex photometrically ambiguous regions. Hybrid approaches—multi-field occupancy/SDF models, local code learning, or explicit uncertainty modeling—mitigate many of these challenges (Lyu et al., 2023, Yao et al., 2021, Dong et al., 2018). Ongoing work seeks to unify explicit and implicit scene representations, extend SDFs to articulated and dynamic objects, and further stabilize neural SDF optimization through geometric and analytic priors (Li et al., 23 Nov 2024, Krishnan et al., 1 Jul 2025). A plausible implication is that future SDF research will increasingly hybridize neural, explicit, and probabilistic components to harness their complementary strengths.