Stochastic Poisson Surface Reconstruction with One Solve using Geometric Gaussian Processes

Poisson Surface Reconstruction is a widely-used algorithm for reconstructing a surface from an oriented point cloud. To facilitate applications where only partial surface information is available, or scanning is performed sequentially, a recent line of work proposes to incorporate uncertainty into the reconstructed surface via Gaussian process models. The resulting algorithms first perform Gaussian process interpolation, then solve a set of volumetric partial differential equations globally in space, resulting in a computationally expensive two-stage procedure. In this work, we apply recently-developed techniques from geometric Gaussian processes to combine interpolation and surface reconstruction into a single stage, requiring only one linear solve per sample. The resulting reconstructed surface samples can be queried locally in space, without the use of problem-dependent volumetric meshes or grids. These capabilities enable one to (a) perform probabilistic collision detection locally around the region of interest, (b) perform ray casting without evaluating points not on the ray's trajectory, and (c) perform next-view planning on a per-ray basis. They also do not requiring one to approximate kernel matrix inverses with diagonal matrices as part of intermediate computations, unlike prior methods. Results show that our approach provides a cleaner, more-principled, and more-flexible stochastic surface reconstruction pipeline.

(1) In computer graphics, recent work on Stochastic Poisson Surface Reconstruction offers a principled way to quantify uncertainty about how to reconstruct a surface from oriented point cloud data, but requires multiple linear solves as part of its computational pipeline. (2) We use ideas from geometric Gaussian processes to reduce this to one linear solve, which scales only with the size of the point cloud data, and not the volumetric grid around it. (3) This produces a computational pipeline for quantifying uncertainty in surface reconstruction which is simpler and more scalable.