Points as Tori: Fast Pointwise Signed Distance for Point Clouds
GraphicsComputer Vision and Pattern Recognition
Summary
The authors present a new method to calculate how far any point in space is from a surface represented by a collection of points (a point cloud). Their approach fits donut-shaped surfaces (tori) locally to the points, which lets them quickly find distances without heavy computations. This method works directly on point clouds from various sources without needing to build a full surface model first. It also connects traditional techniques for understanding shapes with their distance calculations. Their technique supports fast queries, is easy to run in parallel, and can be used for tasks like creating offsets or combining shapes.
Authors
Nicole Feng, Ioannis Gkioulekas, Keenan Crane
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.