Points as Tori: Fast Pointwise Signed Distance for Point Clouds

2026-07-18

AI Summary

This work addresses the challenge of efficiently computing signed distances from arbitrary points to point clouds without explicit surface reconstruction or spatial discretization. It introduces a novel approach that leverages the torus as a local geometric primitive, exploiting its analytic distance function. A pretrained neural network predicts per-point curvature and offset parameters, enabling parallel, mesh-free construction of the signed distance field. The method uniquely unifies signed distance computation with classical reconstruction paradigms—such as winding number and Poisson surface reconstruction—and supports direct geometric operations including offsetting, Boolean combinations, and sphere tracing on point clouds generated from photogrammetry, meshes, 3D Gaussians, or neural implicit representations. This significantly enhances both the efficiency and practicality of downstream geometry processing tasks.

Abstract

We describe a method for computing signed distance to point clouds that allows fast pointwise evaluation at arbitrary spatial resolution. As input, our method takes a point cloud with normals; as output, it provides an analytical parameterization that allows queries of signed distance to the approximate underlying surface at arbitrary points - simultaneously providing reconstruction and distance. Our key idea is to reconstruct shapes by locally fitting point clouds with tori, which have closed-form signed distance functions. Tori are fitted in a feed-forward manner, using a pre-trained network to output per-point curvature and shift parameters. Importantly, our method does not require costly global optimization or spatial discretization, and is easily parallelizable. Underlying our method is a new theory that unifies signed distance with the classic reconstruction methods of winding numbers and Poisson surface reconstruction. We use our method to compute signed distance to point clouds arising from photogrammetry, meshes, 3D Gaussians, and neural implicits. Our method allows point clouds to be used directly in applications, without explicit surface reconstruction: as examples, we take offsets of point clouds, apply morphological and Boolean operations, and directly visualize offset surfaces using sphere tracing.

Problem

• signed distance
• point clouds
• surface reconstruction
• tori
• geometric operations

Innovation

• signed distance function
• point cloud
• torus fitting
• analytical parameterization
• surface reconstruction