2D Surfel Splatting: Methods & Applications

• 2D surfel splatting is a surface-centric representation that uses 2D Gaussian primitives aligned with tangent planes to capture spatial position and orientation.
• It enables accurate real-time scene reconstruction by combining explicit surfel orientation, adaptive scaling, and efficient alpha compositing to manage complex geometries.
• The method is pivotal in SLAM, neural rendering, and semantic mapping, offering enhanced accuracy, computational efficiency, and robust handling of noise.

2D surfel splatting is a surface-oriented scene representation and rendering paradigm in which discrete, oriented geometric primitives—surfels—are parameterized as 2D Gaussian (or disk) elements residing on local tangent planes of the underlying objects. Unlike volume-based rendering or 3D ellipsoidal Gaussians, these surface elements directly encode both spatial position and local orientation, leading to accurate, efficient, and geometrically faithful mapping and visualization. Originating from robotics mapping and computer graphics, 2D surfel splatting has seen a resurgence as a scalable, high-fidelity solution for real-time scene reconstruction in SLAM, multiview stereo, neural rendering, and dense semantic mapping.

1. Mathematical Formulation of 2D Surfel Splatting

Surfel splatting operates by associating each scene point with a local parameterization:

• Center position $p$
• Principal tangents $t_u$, $t_v$ (defining the tangent plane)
• Scales $s_u$, $s_v$ (defining the surfel size along principal axes)
• (In most modern techniques,) Opacity $\alpha$ and a color/appearance feature $c$ or complete texture map

A point on the surfel is parameterized as:

$P(u, v) = p + s_u t_u u + s_v t_v v$

The Gaussian "footprint" on the local plane is:

$G(u, v) = \exp\left(-\frac{u^2 + v^2}{2}\right)$

Rendering proceeds via alpha compositing along viewing rays, sorting surfels in depth order and accumulating their weighted contributions:

$$C(x) = \sum_{i} c_i \omega_i,\quad \omega_i = \alpha_i G_i(u(x)) \prod_{j < i} (1 - \alpha_j G_j(u(x)))$$

where $u(x), v(x)$ are the local coordinates of the image pixel $x$ with respect to the surfel's plane (Fan et al., 28 Jul 2025).

This surface-oriented formulation provides direct normals via $t_w = t_u \times t_v$ and supports efficient intersection evaluation, enabling view-consistent color, depth, and normal map extraction.

2. Core Principles and Benefits of Surface-Aligned Splatting

Contrasting with prior volumetric ("3DGS") schemes, 2D surfel splatting is fundamentally surface-centric:

• Oriented Primitives: Each surfel encodes explicit orientation, delivering more accurate geometry at edges, corners, and thin structures by avoiding the ambiguity of isotropic 3D Gaussian ellipsoids (Fan et al., 28 Jul 2025, Park et al., 2017).
• Computational Efficiency: Projecting and blending Gaussians on 2D tangent planes reduces both the search and compositing space, yielding faster rendering and optimization (Song et al., 2 Dec 2024, Jena et al., 4 May 2025, Fan et al., 28 Jul 2025).
• Scale-Adaptive: Surfel extents adapt to local surface resolution or measurement uncertainty, supporting variable scan density and natural level-of-detail management (Schöps et al., 2018, Park et al., 2017).
• Noise Suppression: By modeling each surfel as a Gaussian density, blending as either Bayesian weighting (Park et al., 2017) or alpha composition, random sensor and estimation noise is attenuated over recursive data fusion or training iterations.

Adaptive data association and fusion strategies, such as dynamic thresholding on spatial proximity and uncertainty (Park et al., 2017), ensure that surfel clouds remain non-redundant, noise-robust, and resolution-adaptive.

3. 2D Surfel Splatting in Mapping, SLAM, and Reconstruction

2D surfel splatting is now a foundation for dense mapping and SLAM systems, enabling real-time, globally consistent scene reconstruction from RGB-D, LiDAR, or multi-view imagery.

• Probabilistic Surfel Fusion: Techniques such as those proposed by (Park et al., 2017) use Bayesian filtering to integrate multiple LiDAR or RGB-D measurements into a dense surfel map. The update rules:

$$\Sigma_{\text{new}} = (\Sigma^{-1} + R^{-1})^{-1},\quad \mu_{\text{new}} = \Sigma_{\text{new}}(\Sigma^{-1}\mu + R^{-1}z)$$

enable principled aggregation of observations, controlling spatial redundancy and per-surfel resolution.

• SLAM with 2DGS Representation: Modern RGB-D SLAM systems (e.g., S³LAM (Fan et al., 28 Jul 2025), 2DGS-SLAM (Zhong et al., 1 Jun 2025)) exclusively utilize 2D Gaussian surfels instead of volumetric primitives. The explicit surface orientation and tangent plane parameterization improve both the quality of the reconstructed map and the convergence basin of camera tracking. Adaptive surface rendering selects between alpha-blended means and dominant surfel values according to local depth distortion, increasing robustness at discontinuities and thin features.
• Mesh and Depth Extraction: Surfel splatting facilitates direct mesh extraction via isosurfacing over high-density surfel clouds or by explicit connectivity updates (cf. SurfelMeshing (Schöps et al., 2018)), supporting rapid updates in the presence of loop closures and pose corrections.
• Sparse and Real-Time Mapping: Recent advances demonstrate that 2D surfel splatting—particularly when initialized using dense MVS or fused stereo point clouds—enables accurate surface reconstruction and novel-view synthesis even from as few as three sparse input images, achieving state-of-the-art results in accuracy and inference time (Wu et al., 29 Apr 2025, Jena et al., 4 May 2025, Takama et al., 26 May 2025).

4. Technical Developments: Depth Consistency, Densification, and Optimization

Addressing the geometric and appearance biases introduced by surfel compositing and view-dependent opacity, contemporary research has introduced several significant innovations:

• Depth Consistency and Correction: To counter artifacts on glossy or specular surfaces—where traditional transmittance-based surface extraction falters—unbiased depth estimation includes "depth convergence loss" and a "rectified depth criterion" (Peng et al., 9 Mar 2025). These enforce continuity and correct surface localization by aggregating weak surfel contributions, mitigating holes in reconstructions.
• Adaptive Densification and Quantile-Based Sampling: In MVG-Splatting (Li et al., 16 Jul 2024), kernel density estimation with FFT acceleration is used to compute adaptive quantile thresholds for identifying under-represented regions in the surfel cloud. A dynamic, geometric-consistency–guided densification adds detail only where needed, thus balancing completeness and efficiency.
• Texture and Shading Models: Augmenting the surfel parameterization with per-surfel texture maps (Song et al., 2 Dec 2024), spherical harmonics-encoded reflection vectors (Jiang et al., 26 Nov 2024), and deferred directional factorization (Zhang et al., 1 Dec 2024) increases fidelity in both diffuse and highly reflective regions. Deferred shading pipelines, spherical Mip-grids, and multi-level grid queries blend spatial and angular factors for photorealistic rendering.
• Efficiency and Pruning: Per-ray sorting for compositing order, in combination with automatic surfel pruning via Fisher information (Song et al., 2 Dec 2024) or cross-view pruning (Xiao et al., 23 May 2025), ensures consistent rendering and scalable memory use at both low and high output resolutions.
• Semantic Surfels and Multi-Task Splatting: Techniques such as efficient semantic splatting (Qi et al., 8 Dec 2024) extend the method to simultaneously render RGB and dense 2D semantic maps using explicit 3D points with learned semantic embeddings, incorporating cross-view and 3D aggregation losses for continuity.

5. Adaptive and Robust Scene Representation: Handling Uncertainty, Thin Structures, and Loop Closure

Surfel splatting is uniquely suited for scenarios requiring adaptivity and robustness:

• Surface-Resolution Adaptation: Surfel size is dynamically assigned based on local measurement gradients or uncertainty. For instance, in (Schöps et al., 2018) the surfel radius is

$$r_s = 1.5 \cdot \max_{u,v \in \{-1,0,1\}} \|p(x, y) - p(x+u, y+v)\|_2$$

thus adjusting to scan density and ensuring seamless "brush stroke" coverage.

• Support for Thin and Non-Manifold Geometry: Surfels can represent non-volumetric structures, including thin objects or double-sided sheets, as their blending is local and does not require enclosing volumes (Schöps et al., 2018, Jiang et al., 26 Nov 2024).
• Asynchronous Remeshing and Loop Closure: In the context of SLAM and online reconstruction, asynchronous remeshing and surfel cloud deformation support efficient, fast adaptation to global corrections, with surfel neighborhoods and positions updated incrementally (Schöps et al., 2018, Fan et al., 28 Jul 2025).

6. Experimental Validation and Applications

Extensive benchmarks across synthetic and real datasets confirm the advantages of 2D surfel splatting:

• Reconstruction Quality: Quantitative metrics such as Chamfer Distance (CD), PSNR, SSIM, and LPIPS demonstrate that surfel-based methods attain or exceed the accuracy of volumetric and implicit approaches, with especially robust performance in sparse-view or loop closure scenarios (Wu et al., 29 Apr 2025, Jena et al., 4 May 2025, Song et al., 2 Dec 2024, Fan et al., 28 Jul 2025).
• SLAM and Robotics: S³LAM (Fan et al., 28 Jul 2025) and 2DGS-SLAM (Zhong et al., 1 Jun 2025) show improved tracking and mapping precision, resilience to viewpoint change, and runtime efficiency suitable for real-time robotics and AR/VR.
• Semantic Segmentation: In remote sensing, surfel-based semantic splatting achieves an order-of-magnitude improvement in inference speed and enhances view/generalization consistency (Qi et al., 8 Dec 2024).

The following table summarizes some major empirical and technical outcomes from recent literature:

Application Area	Technical Innovation	Outcomes/Evidence
RGB-D SLAM	2DGS surfel representation, adaptive rendering, radial Jacobian	State-of-the-art accuracy, high quality mesh, robust tracking (Fan et al., 28 Jul 2025, Zhong et al., 1 Jun 2025)
Sparse Multi-View Reconstruction	MVS/Dense stereo initialization, geometric-prioritized updates	Accurate mesh from 3 views, 2×–100× speedup over implicit methods (Wu et al., 29 Apr 2025, Takama et al., 26 May 2025)
Neural Scene Rendering	Deferred directional factorization, texture-mapped surfels, Mip-grids	Photorealistic NVS, high-frequency detail, real-time rendering (Zhang et al., 1 Dec 2024, Song et al., 2 Dec 2024)
Semantic Mapping	Dual-task RGB/semantic surfels, aggregation losses	Efficient, low-latency multi-view segmentation (Qi et al., 8 Dec 2024)

7. Limitations and Future Directions

Current open challenges and avenues for further research in 2D surfel splatting include:

• Handling of Extremely Sparse or Noisy Regions: Single-view visibility and limited view overlap can leave cracks or geometric holes, particularly in occluded spaces. Addressing this may require new connectivity priors and local continuity constraints (Shen et al., 19 Dec 2024).
• View Consistency and Sorting Artifacts: While per-ray sorting mitigates compositing inconsistencies, computational overhead may be non-negligible. Further algorithmic improvements or hybrid representations (combining surfels with light fields or neural radiance fields) could be explored (Song et al., 2 Dec 2024, Jena et al., 4 May 2025).
• Scalability to Large-Scale Scenes: For unbounded or outdoor scenes, as in modern urban/digital twin applications, hierarchical compensation of tri-plane and local grid features supports both global consistency and high detail (Xiao et al., 23 May 2025), but further optimizations for cross-view synchronization and adaptive sampling remain relevant.
• Integration of External Priors and Semantic Information: Leveraging foundation models (e.g., MASt3R (Zhong et al., 1 Jun 2025), SAM2 (Qi et al., 8 Dec 2024)) for relocalization, pseudo-labeling, or geometry refinement can improve robustness in challenging or label-sparse environments.

A plausible implication is that as 2D surfel splatting frameworks continue to combine geometric, appearance, and semantic information at scale, they will underpin a new generation of efficient, real-time, high-fidelity mapping and rendering systems across robotics, vision, and graphics.