Learning Point Cloud Geometry as a Statistical Manifold: Theory and Practice

TL;DR

This paper introduces a mathematically principled method to model point cloud geometry as a statistical manifold using Gaussian distributions, enabling self-supervised learning and improving robotic perception tasks.

Contribution

The authors propose POLI, a deep neural estimator that predicts local Gaussian geometry in point clouds, removing the need for labeled data and integrating seamlessly into perception pipelines.

Findings

POLI accurately estimates local geometry in point clouds.

The method improves robustness and performance in robotic perception tasks.

Self-supervised learning eliminates the need for labeled data.

Abstract

Point clouds are a fundamental representation for robotic perception tasks such as localization, mapping, and object pose estimation. However, LiDAR-acquired point clouds are inherently sparse and non-uniform, providing incomplete observations of the underlying scene geometry. This makes reliable geometric reasoning challenging and degrades downstream perception performance. Existing approaches attempt to compensate for these limitations by estimating local geometry, but often rely on hand-crafted statistics or end-to-end supervised learning, which can suffer from limited scalability or require large amounts of accurately labeled data. To address these challenges, we explicitly model point cloud geometry under a principled mathematical formulation. We represent local geometry as a statistical manifold induced by a family of Gaussian distributions, where each point is associated with a Gaussian capturing its local geometric structure. Based on this formulation, we introduce Point-to-Ellipsoid (POLI), a deep neural estimator that predicts per-point Gaussian geometry. POLI learns a mapping from point cloud observations to their underlying geometry in a self-supervised manner, removing the need for labeled data while preserving strong geometric inductive biases. The resulting representation integrates seamlessly into existing robotic perception pipelines without architectural modifications. Extensive experiments show that POLI enables accurate and robust geometry estimation and consistently improves performance across diverse robotic perception tasks.