Uncertainty-aware geometry processing on Gaussian Process Implicit Surfaces

We present a framework for uncertainty-aware geometry processing on Gaussian Process Implicit Surfaces (GPIS), enabling computations directly on such probabilistic representations of shapes. In contrast to classical geometry processing pipelines that assume deterministic surface meshes or point clouds, our approach considers uncertainty in the input data and defines analogs of fundamental differential operators-gradient, divergence, and Laplacian- that account for the distribution of plausible geometries encoded by the GPIS. Leveraging the Kac-Rice formula, we embed computations from random surfaces into a volumetric Cartesian domain, enabling efficient evaluation of expected integrals and differential operators. The proposed approach bridges classical surface PDE-based geometry processing and volumetric representations, enabling a principled handling of noise and ambiguity for various downstream geometry processing tasks.