EdgeGaussians: 3D Edge Mapping

• The paper introduces EdgeGaussians, an explicit 3D edge mapping method that models edge points as 3D Gaussians to achieve rapid convergence and high geometric fidelity.
• It employs differentiable rendering and edge-aware loss functions to align projected Gaussians with semantic 2D edge maps, improving edge detection and reconstruction.
• The approach reduces training time significantly while enhancing scene understanding applications like SLAM, CAD modeling, and high-fidelity rendering.

Edges form the backbone of geometric abstraction in computer vision, providing compact and structurally significant representations for mapping, segmentation, SLAM, and high-fidelity rendering. The field of 3D edge mapping merges advances in neural fields, explicit geometric primitives, and fast differentiable rendering—culminating in the EdgeGaussians approach, which unifies these under the 3D Gaussian Splatting (3DGS) framework. This article presents the principles, methodologies, and implications of EdgeGaussians and related edge-based 3D mapping techniques, with precise technical details reflecting recent literature.

1. Edges as Core Primitives in 3D Scene Representation

3D edges, consisting of lines and curves, distill the essential geometric structure of objects and scenes. Unlike surfaces, edges are less dependent on textures or photometric diversity, serving as robust features for reconstruction under a variety of input modalities and environmental conditions (Chelani et al., 19 Sep 2024). Edge-centric representations excel in applications such as CAD modeling, scene understanding, localization, SLAM, and mapping—enabling efficient abstraction and structural fidelity.

Traditional methods abstract lines from multi-view geometry (e.g., L3D++, LIMAP), but tend to overlook curves and suffer from redundancy, fragmentation, and noisy point clouds (Yang et al., 30 Nov 2024, Li et al., 29 May 2024). Neural methods leveraging implicit fields (SDF/UDF) encode edges as loci within learned scalar or probability fields, but face challenges with sampling precision and heavy computational resources.

2. Explicit Edge Representation: EdgeGaussians Methodology

EdgeGaussians departs from implicit field approaches, providing an explicit, parametric representation of 3D edges within a 3DGS framework (Chelani et al., 19 Sep 2024). Each 3D edge point is modeled as a 3D Gaussian with:

• Mean ($\mu \in \mathbb{R}^3$): Spatial location of the edge point.
• Covariance ($\Sigma \in \mathbb{R}^{3 \times 3}$): Encodes both spatial uncertainty and the local edge direction as the principal axis (largest eigenvector).
• Opacity ($\alpha$): Determines edge salience in rendering.

Edge points are learned directly from multi-view images. Supervision utilizes 2D edge maps (from PiDiNet, DexiNed, or similar detectors), enforcing alignment of the projected 3D Gaussians to these semantic edges. The masked L1 loss targets both edge and non-edge pixels for balanced optimization:

$$\mathcal{L}_{\text{proj}} = \text{mean}\left( \mathcal{M} \odot | \hat{I} - I| \right)$$

where $\mathcal{M}$ is a mask selecting all edge pixels and an equal set of non-edge pixels. The total loss includes direction consistency and shape regularization:

$$\mathcal{L} = \mathcal{L}_{\text{proj}} + \lambda_1 \mathcal{L}_{\text{orient}} + \lambda_2 \mathcal{L}_{\text{shape}}$$

$$\mathcal{L}_{\text{orient}} = 1 - \frac{1}{N}\sum_{i=1}^N\frac{1}{k} \sum_{j=1}^k |d_i^T d_{i_j}|$$

$$\mathcal{L}_{\text{shape}} = \frac{1}{N} \sum_{i=1}^{N} \frac{s_i^{(2)}}{s_i^{(1)}}$$

Model parameters are jointly optimized via gradient descent, providing convergence within minutes versus hours for implicit field methods.

3. Gaussian Splatting and Edge-Focused Rendering

The EdgeGaussians framework leverages the computational and representational efficiency of 3D Gaussian Splatting (Chelani et al., 19 Sep 2024, Huang et al., 6 Aug 2025). Scenes are rendered by projecting anisotropic 3D Gaussians to the image plane, aggregating alpha-blended contributions per pixel.

Recent advancements, such as 3DGEER (Huang et al., 29 May 2025), introduce mathematically exact closed-form volumetric integration of density along rays and the Particle Bounding Frustum (PBF) for ray-Gaussian association, improving edge fidelity and computational performance under wide-FoV and distorted camera models. This mathematically rigorous approach prevents edge smearing and artifacts common in projective approximations, yielding crisp and robust edge mapping.

$$T(\mathbf{o}, \mathbf{d}) = \sigma \exp\left( -\frac{1}{2} \mathrm{D}_{\boldsymbol{\mu}, \Sigma}(\mathbf{o}, \mathbf{d})^2 \right)$$

where $$\mathrm{D}_{\boldsymbol{\mu}, \Sigma}(\mathbf{o}, \mathbf{d})$$ is the minimal Mahalanobis distance from the ray to the Gaussian center.

4. Edge Detection and Regularization within Neural Fields

Edge mapping can also be approached via density-gradient techniques on NeRF-based scenes (Jäger et al., 2023). Filtering the voxelized density field with 3D Sobel, Canny, or LoG operators enables extraction of 3D edges as isosurfaces of maximal gradient magnitude, independent of absolute density scaling. The 3D Canny filter, in particular, produces gapless, uniformly sampled point clouds with favorable completeness/accuracy trade-offs:

$$\Delta_{\delta, \text{Sobel}} = \sqrt{G_{\delta,x}^2 + G_{\delta,y}^2 + G_{\delta,z}^2}$$

This approach acts as a bridge from implicit NeRF representations to explicit edge point clouds, supporting downstream edge parameterization.

Regularization of edge features in 3DGS pipelines is critical for geometric integrity. DET-GS (Huang et al., 6 Aug 2025) incorporates semantic edge-aware masking (via Canny detection) and RGB-guided TV loss to constrain smoothing, rigorously preserving scene boundaries and textures:

$$\mathcal{L}_{edge} = \frac{1}{P} \sum_{x_i} m(x_i) \cdot | \mathcal{D}(x_i) - \overline{\mathcal{D}(x_i)} |^2$$

$$\mathcal{L}_{tv} = \frac{1}{|\Omega|} \sum_{x \in \Omega} \left( M_h(x) \cdot \max\{ |\nabla_h \mathcal{I}_{pred}(x)| - \tau_{smooth}, 0 \} + M_v(x) \cdot \max\{ |\nabla_v \mathcal{I}_{pred}(x)| - \tau_{smooth}, 0 \} \right)$$

A plausible implication is that these regularization strategies are necessary for robust edge mapping under sparse-view conditions or challenging image domains.

5. Edge Extraction, Model Fitting, and Scene Abstraction

Once Gaussian parameters for edge points are inferred, forming parametric edge curves in 3D requires clustering and fit refinement. EdgeGaussians employs simple graph traversal based on spatial proximity and directional alignment to connect edge points into chains. For each cluster, both 3D lines and Bezier curves are fit, with the minimum residual chosen to encode the edge.

LineGS (Yang et al., 30 Nov 2024) refines geometry-guided initial line segments by exploiting high-density Gaussian distributions along scene boundaries. Cylinder-based mapping finds supporting Gaussians for each segment, which are then used for position correction (linear regression), overextension cropping (binary search based on density falloff), and duplication/discontinuity elimination (segment clustering and merging).

$$X(\vec{s}) = \{ x \in G \mid x \in C(\vec{s}, r) \}$$

$$\mathbf{s}(\vec{s}_i, \vec{s}_j) = \frac{\tanh(R^2 \cdot \cos\theta)}{1 + \lambda \cdot d^2(\vec{s}_i, \vec{s}_j)}$$

These techniques yield compact and structurally faithful line-based abstractions of complex scenes.

6. Quantitative Evaluation and Practical Implications

EdgeGaussians and related methods are quantitatively benchmarked on datasets such as ABC-NEF (parametric CAD), DTU (real scenes), and complex indoor/outdoor scenes. EdgeGaussians achieves geometric accuracy and completeness on par or superior to implicit neural field baselines (NEF, EMAP), especially at moderate thresholds, and with up to 30$\times$ lower training time (Chelani et al., 19 Sep 2024).

DET-GS and edge-aware regularization methods yield consistent improvements in PSNR, SSIM, and LPIPS, surpassing SOTA baselines in precision, especially for thin structures and scene boundaries (Huang et al., 6 Aug 2025).

LineGS demonstrates reductions in RMSE and over-segmentation, with model abstraction scores climbing up to 43% in complex environments (Yang et al., 30 Nov 2024).

EGGS (Gong, 14 Apr 2024), via edge-weighted loss, improves rendering sharpness and PSNR by 1–2 dB across multiple datasets without increasing compute cost.

7. Connections to Broader Research and Applications

The convergence of explicit, Gaussian-based edge mapping and neural field techniques establishes new directions for 3D perception and abstraction. EdgeGaussians unifies the strengths of geometric precision, rapid optimization, and differentiable rendering, providing scalable solutions for SLAM, semantic mapping, CAD model generation, and advanced rendering pipelines.

EMAP (Li et al., 29 May 2024) and other UDF-based methods implicate the utility of neural distance priors not only for edge reconstruction but as powerful initiators for surface meshing. The field is ripe for integration between learned edge priors and Gaussian-based surface modeling, potentially facilitating topologically faithful and physically consistent scene reconstruction.

Table: Edge Representation Methods — Comparison

Method	Representation	Training Time	Accuracy/Completeness
NEF/EMAP	Implicit Field	Hours	SOTA; strict threshold pref
EdgeGaussians	Explicit Gaussians	Minutes	On-par or better most metrics
LineGS	Geometry + GS Model	Variable	Higher abstraction/precision
DET-GS	Edge-reg. 3DGS	GS time	Improved detail/boundaries
EGGS	Edge-weighted loss	GS time	1–2 dB PSNR gain

This illustrates the transition from implicit, resource-intensive networks to rapid, explicit, and geometrically interpretable edge mapping methods.

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EdgeGaussians and affiliated edge mapping strategies leverage advances in Gaussian splatting, neural fields, and differentiable rendering for accurate, efficient, and interpretable 3D edge reconstruction. The field exhibits significant progress in scalability and structural fidelity, with ongoing research exploring integration with broader geometric and neural representations for applications in scene modeling, robotics, and computer graphics.