Stochastic Poisson Surface Reconstruction with One Solve using Geometric Gaussian Processes

2025-03-24

AI Summary

Poisson surface reconstruction from partial or sequential point clouds using Gaussian processes (GPs) traditionally involves a two-stage pipeline—first interpolating point-wise normals via GP regression, then solving the volumetric Poisson PDE globally—entailing high computational cost and reliance on diagonal approximations of the kernel matrix inverse. Method: We propose a unified, single-stage framework that embeds geometric Gaussian processes directly into the Poisson reconstruction formulation, jointly modeling surface geometry and uncertainty. This integrates normal interpolation and PDE solving into one sparse linear system, enabling mesh-free function evaluation and local spatial reasoning—including collision detection, on-demand ray casting, and slice-level view planning—without explicit kernel matrix operations. Results: Experiments demonstrate superior reconstruction accuracy and real-time performance over conventional two-stage approaches, validating its efficacy for task-driven, online 3D perception.

Technology Category

• Knowledge Representation and Reasoning: Geometric, Spatial, and Temporal Reasoning
• Planning, Routing, and Scheduling: Replanning and Plan Repair
• Computer Vision: 3D Computer Vision

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Data management and stream processing for Web, mobile and wireless applications
• Economics, Online Markets and Human Computation: Economic ramifications for generative AI infrastructure and applications

Abstract

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-slice basis. They also improve reconstruction quality, by not requiring one to approximate kernel matrix inverses with diagonal matrices as part of intermediate computations. Results show that our approach provides a cleaner, more-principled, and more-flexible stochastic surface reconstruction pipeline.

Problem

• Combines interpolation and surface reconstruction into one stage
• Enables local surface queries without volumetric meshes
• Improves reconstruction quality by avoiding diagonal matrix approximations

Innovation

• Combines interpolation and reconstruction in one stage
• Uses geometric Gaussian processes for single linear solve
• Enables local querying without volumetric meshes