One-Shot Method for Computing Generalized Winding Numbers

TL;DR

This paper introduces a universal, efficient method for computing generalized winding numbers using surface boundaries and ray intersections, applicable to various surface representations and faster than existing methods in many cases.

Contribution

The authors present a novel one-shot algorithm for generalized winding number computation that supports any oriented surface with ray intersection queries, improving speed and flexibility.

Findings

Up to 4x faster than state-of-the-art for 2D curves on regular grids.

Capable of computing winding numbers for parametric surfaces without discretization.

Faster than hierarchical and adaptive methods for certain meshes and parametric surfaces.

Abstract

The generalized winding number is an essential part of the geometry processing toolkit, allowing to quantify how much a given point is inside a surface, even when the surface has boundaries and noise. We propose a new universal method to compute a generalized winding number, based only on the surface boundary and the intersections of a single ray with the surface, supporting any oriented surface representations that support a ray intersection query. Due to the focus on the boundary, our algorithm has a unique set of properties. For 2D parametric curves, on a regular grid of query points, our method is up to 4x faster than the current state of the art, maintaining the same precision. In 3D, our method can compute a winding number of a surface without discretizing it, including parametric surfaces. For some meshes with many triangles and a simple boundary, our method is faster than the hierarchical evaluation of the generalized winding number while still being precise. Similarly, on some parametric surfaces with a simple boundary, our method can be faster than adaptive quadrature. We validate our algorithms theoretically, numerically, and by demonstrating a gallery of results on a variety of parametric surfaces and meshes, as well uses in a variety of applications, including voxelizations and boolean operations.