A Probabilistic Approach to Uncertainty Quantification Leveraging 3D Geometry

Quantifying uncertainty in neural implicit 3D representations, particularly those utilizing Signed Distance Functions (SDFs), remains a substantial challenge due to computational inefficiencies, scalability issues, and geometric inconsistencies. Existing methods typically neglect direct geometric integration, leading to poorly calibrated uncertainty maps. We introduce BayesSDF, a novel probabilistic framework for uncertainty quantification in neural implicit SDF models, motivated by scientific simulation applications with 3D environments (e.g., forests) such as modeling fluid flow through forests, where precise surface geometry and awareness of fidelity surface geometric uncertainty are essential. Unlike radiance-based models such as NeRF or 3D Gaussian splatting, which lack explicit surface formulations, SDFs define continuous and differentiable geometry, making them better suited for physical modeling and analysis. BayesSDF leverages a Laplace approximation to quantify local surface instability via Hessian-based metrics, enabling computationally efficient, surface-aware uncertainty estimation. Our method shows that uncertainty predictions correspond closely with poorly reconstructed geometry, providing actionable confidence measures for downstream use. Extensive evaluations on synthetic and real-world datasets demonstrate that BayesSDF outperforms existing methods in both calibration and geometric consistency, establishing a strong foundation for uncertainty-aware 3D scene reconstruction, simulation, and robotic decision-making.