Geometry-Grounded Gaussian Splatting

TL;DR

The paper addresses the challenge of reliable shape extraction from Gaussian Splatting by introducing Geometry-Grounded Gaussian Splatting, which treats Gaussian primitives as stochastic solids to enable principled depth rendering and explicit geometry extraction. It proves the rendering equivalence between Gaussian primitives and stochastic solids and derives a vacancy-based attenuation model, including a median-depth definition and closed-form depth gradients for efficient training. The approach yields state-of-the-art geometry reconstruction among Gaussian Splatting methods on public datasets, with strong multi-view consistency and cleaner depth maps that facilitate high-fidelity mesh extraction. This framework broadens Gaussian Splatting to principled geometry optimization and offers a pathway to fully volumetric rendering in the future, potentially enhancing practical applications in 3D reconstruction and view synthesis.

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.

Figures (14)

• Figure 1: Overview of our depth-rendering pipeline. (a) We rasterize Gaussian primitives and sort them by depth. (b) Standard Gaussian Splatting yields step-wise transmittance under splat compositing. (c) Under our stochastic solid formulation, attenuation is modeled continuously within each primitive, yielding a smooth transmittance curve. (d) Prior work estimates the ray-wise median depth as the point where transmittance drops to 0.5. (e) Forward pass: we locate the median depth $t_{med}$, i.e., $T=0.5$, via binary search. (f) Backward pass: we backpropagate through $t_{med}$ using a closed-form gradient with respect to all Gaussians contributing to the ray.
• Figure 2: Given a Gaussian primitive, we regard it as a stochastic solid and derive an appropriate attenuation coefficient with Equation \ref{['eq:oav']}. With our attenuation coefficient, the volume rendering of this stochastic solid is equivalent to the rasterization rendering developed in the original Gaussian Splatting.
• Figure 3: Depth maps of Gaussian Splatting-based methods. We visualize the depth maps by converting them to 3D points. Our method produces a clean and smooth depth map. PGSR uses expected depth, yielding much noise at edges. GOF uses median depth and suffers from unsmooth depth changes.
• Figure 4: Illustration of depth rendering methods. (a) The green plane’s opacity $\alpha$ decreases smoothly from 1 on the right to 0 on the left. Consequently, (b) the median depth changes in a step-like manner, whereas (c) the expected depth varies continuously.
• Figure 5: Illustration of vacancy along the camera ray. The vacancy value on the ray equals the transmittance on the front side of the Gaussians.