Focused Gaussian Surfel Splatting

• The paper introduces a focused Gaussian surfel splatting approach that optimizes anisotropic 3D surfel primitives for maximal surface fidelity and photorealistic rendering.
• It employs adaptive initialization and density control to concentrate computational resources on geometrically salient regions, ensuring multi-view consistency.
• The method integrates differentiable rendering with global pose optimization and GPU-accelerated pipelines, enabling robust SLAM, point cloud completion, and large-scale mapping.

Focused Gaussian Surfel Splatting is a class of techniques that constructs, renders, and optimizes 2D-oriented Gaussian primitives ("surfels") in 3D space, with algorithmic mechanisms for concentrating representation and computation at geometrically salient or data-rich regions. The objective is to achieve maximal surface fidelity, multi-view consistency, and photorealistic rendering with strong geometric alignment, especially under conditions of sparse input, sensor fusion, or real-time requirements. Modern systems leverage explicit geometric constraints, data-driven surfel initialization, scale control, and hierarchical optimization to ensure that the Gaussian surfel representation is both efficient and "focused" on surface regions critical for reconstruction quality and downstream tasks (Park et al., 31 Jul 2025, Strobel et al., 2 Dec 2025, Pan et al., 1 Dec 2025, Fan et al., 28 Jul 2025).

1. Surfel Representation and Splatting Formulation

A Gaussian surfel is defined as an anisotropic ellipsoidal or disk-like primitive in 3D, parametrized minimally by centroid $\boldsymbol{\mu}\in\mathbb{R}^3$, principal directions (tangents and normal), principal axis scales $(s_u, s_v)$, and potentially opacity weight $w$, surface normal $\mathbf{n}$, and a color model, typically encoded as spherical harmonics for view-dependent appearance (Strobel et al., 2 Dec 2025, Park et al., 31 Jul 2025, Fan et al., 28 Jul 2025). The 3D density function is: $$g(\mathbf{x}) = w\,\mathcal{N}(\mathbf{x}\mid\boldsymbol{\mu},\boldsymbol{\Sigma}), \quad \boldsymbol{\Sigma} = s_u^2\,\mathbf{t}_u\mathbf{t}_u^\top + s_v^2\,\mathbf{t}_v\mathbf{t}_v^\top + \varepsilon\,\mathbf{n}\mathbf{n}^\top,$$ where $\varepsilon$ is a small regularization parameter for thickness. Once projected into the camera, each surfel induces a 2D elliptical Gaussian "splat," with image-plane covariance obtained via the linearized projection Jacobian: $$\boldsymbol{\Sigma}_{2D} = \mathbf{J}\,\boldsymbol{\Sigma}\,\mathbf{J}^\top,$$ where $\mathbf{J}$ encodes the local derivatives given camera intrinsics and pose (Park et al., 31 Jul 2025, Pan et al., 1 Dec 2025).

Rendered color, depth, and normal at pixel $\mathbf{u}$ are typically computed by front-to-back $\alpha$-compositing: $$C(\mathbf{u}) = \sum_{i=1}^N T_i(\mathbf{u})\,\alpha_i(\mathbf{u})\,c_i, \quad D(\mathbf{u}) = \frac{\sum_{i=1}^N T_i(\mathbf{u})\,\alpha_i(\mathbf{u})\,z_i(\mathbf{u})}{\sum_{i=1}^N T_i(\mathbf{u})\,\alpha_i(\mathbf{u})}, \quad T_0=1,\,T_i=\prod_{j=0}^{i-1}[1-\alpha_j(\mathbf{u})]$$ where $\alpha_i(\mathbf{u})$ is the per-pixel weight generated by the projected 2D Gaussian (Pan et al., 1 Dec 2025, Fan et al., 28 Jul 2025).

2. Focused Initialization and Adaptive Density Control

Central to the "focused" concept is the adaptive placement and shape initialization of surfels so that computational density aligns with geometric and appearance complexity:

• Pixel-aware Initialization: Starting from a LiDAR or photogrammetric point cloud, local tangent orientation is estimated via PCA or local fitting. Surfel covariances are initialized anisotropically—large in the tangent plane, thin along the normal—then scaled to target a desired image-plane footprint, computed by projecting through the camera model and adjusting by the determinant ratio for Gaussian image coverage (Park et al., 31 Jul 2025).
• Region-focused Generation: In online/run-time mapping, surfels are added only in regions where existing surfel coverage is insufficient, as evidenced by opacity accumulation tests or foreground emergence. New surfels are spawned directly at RGB-D depth discontinuities or where current $\alpha$-composite coverage falls below threshold, with adaptive scale proportional to scene depth (Pan et al., 1 Dec 2025).
• Ambiguity Heuristics in Completion: SurfFill identifies high-ambiguity or under-scanned regions in LiDAR via local neighbor distance statistics; focused surfel growth is constrained to these transition zones to avoid unnecessary over-completion (Strobel et al., 2 Dec 2025).
• Scale Constraint and Bounded Growth: For robustness, surfel scale parameters are constrained using a bounded-sigmoid mapping,

$$\sigma_{\rm bounded} = \sigma_{\min} + (\sigma_{\max} - \sigma_{\min})\,\mathrm{sigmoid}(s),$$

and dynamically regulated to prevent uncontrolled growth in sparse or ambiguous areas (Park et al., 31 Jul 2025).

3. Differentiable Rendering, Optimization, and Global Alignment

Focused Gaussian surfel systems leverage end-to-end differentiable pipelines for both geometric and appearance optimization:

• SLAM/Mapping Back-ends: Systems such as GSFusion and S$^3$LAM incorporate real-time pipelines alternating between local windowed surfel and pose optimization (with photometric, geometric, and ICP-style residuals) and global pose-graph refinement incorporating surfel-to-surfel alignment costs: $$e_{\mathrm{surfel}}(i, j) = \mathbf{d}^\top(\Sigma_j + {}^i\mathbf{R}_j\,\Sigma_i\,{}^i\mathbf{R}_j^\top)^{-1}\mathbf{d},$$ where $\mathbf{d}$ encodes the transformed surfel center difference between frames (Park et al., 31 Jul 2025).
• Surface-focusing Blending: To avoid "blurring" at edges and to sharpen geometry with limited surfel budgets, S$^3$LAM and related methods compute a per-pixel depth-distortion

$$\mathcal{D}_d(\mathbf{x}) = \sum_{i,j} w_i w_j |z_i - z_j|,$$

enabling selective replacement of weighted averages by the contribution of the dominant surfel in ambiguous pixels (Fan et al., 28 Jul 2025).

• Analytical Pose Jacobians: Accurate SLAM and mapping leverage analytical SE(3) derivatives, including both tangential and "radial" components, directly computed from the surfel intersection point—the inclusion of this extra radial term in the Jacobian is unique to 2D surfel splatting and widens the convergence basin compared to isotropic 3D Gaussians (Fan et al., 28 Jul 2025).
• Region-weighted Losses and Regularization: Focused optimization pipelines weight color, edge, and smoothness losses preferentially in ambiguous or data-rich areas, with additional depth and normal consistency terms applied globally to prevent degenerate surfel growth or collapse (Strobel et al., 2 Dec 2025).

4. Large-scale Systems and Real-time Performance

Scalability and real-time throughput are addressed via several architectural and algorithmic innovations:

• Divide-and-conquer Optimization: To process building-scale datasets (e.g., $\sim$70M points) within practical runtimes, the domain is spatially chunked with overlapping tiles; each chunk is processed independently, and boundary continuity is enforced through marginal overlap (Strobel et al., 2 Dec 2025).
• Data Structures and GPU Pipelines: GPU-accelerated rasterization enables high frame-rate mapping and rendering. Two-tier map representations (lightweight surfel map for odometry, dense map for rendering) support efficient access and update (Park et al., 31 Jul 2025).
• Focused Splatter Management: Surfel addition and deletion, densification through split/clone criteria, and selective pruning are implemented as highly parallel GPU operations, often gated by contribution measures and running-average error (Fan et al., 28 Jul 2025).
• Adaptive foveated rasterization and visibility culling: In large-FOV and VR applications, tile-based adaptive splatting with per-tile visibility and sorting ensures consistent rendering with minimal artifacts and optimal computation in focus regions (Tu et al., 15 May 2025).

5. Applications, Benchmarks, and Limitations

Focused Gaussian surfel splatting underpins or extends a variety of high-impact applications:

• Simultaneous Localization and Mapping (SLAM): GSFusion achieves online LiDAR-inertial-visual mapping with sub-pixel global alignment and real-time rendering, surpassing prior 3DGS SLAM systems in efficiency and map quality (Park et al., 31 Jul 2025).
• Point Cloud Completion: SurfFill leverages ambiguity heuristics to complete LiDAR point clouds, outperforming prior LiDAR, MVS, and NeRF-based methods in recall and precision for thin or ambiguous structures (Strobel et al., 2 Dec 2025).
• Real-time RGB-D Mapping: S$^3$LAM achieves highly accurate geometric tracking and mapping with a focused, 2D surfel representation, including analytical tracking Jacobians and dynamic edge sharpening (Fan et al., 28 Jul 2025).
• Photorealistic Scene Reconstruction: Transformer-based approaches such as FSFSplatter, with dense self-splitting initialization and multi-term, differentiable losses, achieve rapid and accurate 3-view reconstructions (Zhao et al., 3 Oct 2025).
• Sensor Fusion and Uncertainty Modeling: EGG-Fusion fuses RGB-D observations into a surfel map via an information filter, explicitly tracking surfel uncertainty in position and orientation for robust real-time operation (Pan et al., 1 Dec 2025).

Limitations:

• Coverage gaps may persist in single-view or sparsely-observed regions (e.g., cracks or overgrowth in SolidGS, SurfFill).
• Blockwise optimization and merging can introduce boundary artifacts if overlap/continuity are not adequately managed (cf. GaussianFocus).
• Highly specular or transparent surfaces, and deeply concave interiors, may not be reliably modeled in the current Gaussian surfel parameterization.

6. Quantitative Results and Performance

Empirical evaluations consistently indicate that focused surfel splatting delivers superior geometric and photometric performance:

Method / Dataset	Chamfer (mm) ↓	PSNR (dB) ↑	SSIM ↑	LPIPS ↓	Runtime	Reference
SolidGS (DTU, 3-view)	1.27	21.32	—	—	3.1 min	(Shen et al., 19 Dec 2024)
SurfFill (synthetic avg)	0.0055	—	—	—	25 min/chunk	(Strobel et al., 2 Dec 2025)
FSFSplatter (DTU)	1.58	30.07	0.906	0.113	<3 min total	(Zhao et al., 3 Oct 2025)
GSFusion (KITTI)	—	—	—	—	Real-time	(Park et al., 31 Jul 2025)

Experiments also demonstrate qualitative superiority in edge preservation, suppression of floaters, and generalization to novel views under sparse supervision.

7. Perspectives and Future Directions

Several research trajectories are apparent:

• Automated or adaptive block merging, attention sparsification, and overlap-aware fusion to scale subdivision-based pipelines to even larger or more complex scenes (Huang et al., 22 Mar 2025).
• Explicit modeling of radiance and BRDF with learned surface properties to capture appearance generalization and support non-Lambertian effects.
• Extension to dynamic scenes via time- or motion-aware surfel parameters, supporting SLAM or reconstruction in changing environments.
• Hybridization with mesh or implicit representations for robust single-view or occluded region completion.

A plausible implication is that focused Gaussian surfel splatting, with explicit attention to geometric significance and uncertainty, will become foundational for robust, real-time, high-fidelity 3D reconstruction and mapping in diverse sensing settings.