Geometry-Grounded Gaussian Splatting

Abstract: Gaussian Splatting (GS) has demonstrated impressive quality and efficiency in novel view synthesis. However, shape extraction from Gaussian primitives remains an open problem. Due to inadequate geometry parameterization and approximation, existing shape reconstruction methods suffer from poor multi-view consistency and are sensitive to floaters. In this paper, we present a rigorous theoretical derivation that establishes Gaussian primitives as a specific type of stochastic solids. This theoretical framework provides a principled foundation for Geometry-Grounded Gaussian Splatting by enabling the direct treatment of Gaussian primitives as explicit geometric representations. Using the volumetric nature of stochastic solids, our method efficiently renders high-quality depth maps for fine-grained geometry extraction. Experiments show that our method achieves the best shape reconstruction results among all Gaussian Splatting-based methods on public datasets.

• The paper formalizes Gaussian primitives as stochastic solids to provide a robust analytic attenuation coefficient for volume rendering and precise geometric isosurface definition.
• It employs a novel depth rendering pipeline using median depth computation and binary search, yielding smooth and multi-view consistent depth maps.
• Experimental results on DTU and Tanks & Temples benchmarks demonstrate superior shape reconstruction accuracy and reduced runtimes compared to existing methods.

Geometry-Grounded Gaussian Splatting: A Technical Summary

Motivation and Theoretical Foundation

The work "Geometry-Grounded Gaussian Splatting" (2601.17835) addresses a fundamental limitation of Gaussian Splatting (GS) for multi-view 3D reconstruction: the absence of an intrinsic geometric definition in Gaussian primitives. While GS yields high-quality and efficient novel view synthesis via rasterized 3D Gaussians, existing shape reconstruction approaches are primarily heuristic, leading to artifacts such as floaters and inconsistent geometry across views.

This paper bridges GS and geometry-grounded radiance field methods (e.g., NeRF variants) by establishing a rigorous theoretical framework: Gaussian primitives are formalized as stochastic solids following the stochastic geometric interpretation introduced in "Objects as Volumes" [Miller:VOS:2024]. This connection allows GS to inherit principled volumetric rendering properties and robust geometry parameterization.

Figure 1: Overview of the depth-rendering pipeline, contrasting standard GS with the proposed stochastic solid formulation; the latter yields continuous transmittance and smooth geometric isosurfaces.

Gaussian Primitives as Stochastic Solids

The main technical contribution is the derivation of an explicit attenuation coefficient $\sigma$ for Gaussian primitives, based on their occupancy and vacancy fields. Specifically, if $G(\mathbf{x})$ is the Gaussian kernel, the vacancy is defined as $\mathrm{v}(\mathbf{x}) = \sqrt{1-G(\mathbf{x})}$, ensuring differentiability and correct monotonicity. This allows the computation of the attenuation coefficient:

$$\sigma(\mathbf{x},\mathbf{\omega}) = \big| \mathbf{\omega} \cdot \nabla \log(\mathrm{v}(\mathbf{x})) \big|$$

Resultantly, volume rendering of a Gaussian solid produces the same compositional result as GS rasterization under closed-form integration. This equivalence moves geometric readouts away from pixel-wise heuristics and toward robust, analytic isosurface definitions.

Figure 2: Illustration of a single Gaussian primitive as a stochastic solid, with attenuation derived analytically and volume rendering matching rasterization.

Depth Rendering and Multi-View Consistency

The stochastic solid formulation enables the computation of smooth depth maps with high multi-view consistency. Instead of alpha-weighted or plane-based depth heuristics, depth is analytically defined as the median along a ray where the accumulated transmittance drops to a threshold ($T=0.5$). This crossing is found efficiently via binary search leveraging monotonicity.

Implementation benefits include:

• Continuous and differentiable transmittance: enables smooth optimization and gradient flow to all contributing Gaussians.
• Efficient backpropagation: a closed-form solution for the gradient of median depth, allowing distributed update to all Gaussians on a ray.

  Figure 3: Depth maps converted to 3D points; the proposed method (center) yields clean, smooth, edge-preserving depth maps, while prior GS methods suffer from noisy boundaries and cross-view inconsistency.

  

  Figure 4: Green plane experiment showing median and expected depth variation: the stochastic solid approach produces step-free median isosurfaces and avoids view-dependent artifacts.

  

  Figure 5: Visualizing vacancy and transmittance along the camera ray—they coincide on the front side of high-opacity Gaussians, yielding robust geometric classification.

Experimental Results

Quantitative evaluation on DTU and Tanks & Temples benchmarks demonstrates that the geometry-grounded GS method achieves the best shape reconstruction among all explicit GS-based methods, rivaling state-of-the-art volumetric approaches while maintaining GS's acceleration and scalability.

• On DTU: Chamfer Distance matches GeoSVR, but at significantly reduced runtime (15 vs. 53 minutes for comparable iterations).
• On Tanks & Temples: F1-score surpasses prior GS baselines, robustly capturing fine details and correctly handling floaters via analytic isosurfaces.

Qualitative comparisons further show cleaner mesh extraction, sharper silhouettes, and improved detail recovery.

Figure 6: Qualitative comparison on Tanks & Temples: the proposed method reconstructs plausible meshes with superior preservation of fine geometry compared to PGSR and GeoSVR.

Figure 7: Cycle reprojection error during optimization; the geometry-grounded approach converges faster and achieves lower foreground error, evidencing stronger multi-view consistency.

Ablation and Further Analysis

The proposed analytic depth computation provides stronger geometric supervision for optimization, outperforming previous regularizers based on normal consistency and exposure compensation. Multi-view regularization plays a reduced role when depth is analytically grounded, demonstrating the robustness of the intrinsic geometric field.

Figure 8: Depth rendering comparison in 3D point cloud space; only the stochastic-solid approach yields consistent and sharp surfaces.

Figure 9: Projection error comparison with PGSR and RaDe-GS baseline: only the analytic approach eliminates floaters and snaps to true surface correspondences across views.

Implications and Future Directions

This work reconciles the efficiency–accuracy tradeoff in photometric 3D reconstruction by endowing GS with explicit, robust geometric fields. Potential future research avenues include:

• Extending stochastic volume rendering to color and normal maps: currently, only depth exploits full volumetric effects for efficiency.
• Adaptive search intervals in large-scale scenes: tightening initial bracketing for median-depth computation could further accelerate optimization.
• Hybrid mesh extraction pipelines: integrating tetrahedralization strategies that leverage Gaussian-specific distribution properties to better recover thin structures.
• Cross-pollination with NeRF geometric regularization: utilizing advanced priors from implicit surface methods within GS for further detail enhancement.

  Figure 10: Results on Mip-NeRF 360: geometry-grounded GS can be extended to novel view synthesis tasks, with spherical Gaussian mixtures improving specular rendering.

Conclusion

Geometry-Grounded Gaussian Splatting formalizes the connection between Gaussian rasterization and volumetric stochastic solids in radiance field reconstruction. By deriving analytic attenuation and isosurface definitions for Gaussians, the method provides efficient, consistent, and accurate geometry extraction directly from GS primitives, setting a new standard for shape fidelity and optimization speed among explicit scene representations.