A Connected Component Labeling Algorithm for Implicitly-Defined Domains

TL;DR

This paper introduces a recursive algorithm for labeling connected components in domains defined by multivariate polynomials, with guarantees of correctness and robustness in complex geometric scenarios.

Contribution

The paper presents a novel recursive subdivision algorithm that uses Bernstein polynomial properties to accurately identify connected components in implicitly-defined domains.

Findings

Algorithm effectively handles complex geometries including cusps and self-intersections.

Provides certificates of correctness for component labeling.

Performs well in statistical tests on random and specific geometric examples.

Abstract

A connected component labeling algorithm is developed for implicitly-defined domains specified by multivariate polynomials. The algorithm operates by recursively subdividing the constraint domain into hyperrectangular subcells until the topology thereon is sufficiently simple; in particular, we devise a topology test using properties of Bernstein polynomials. In many cases the algorithm produces a certificate guaranteeing its correctness, i.e., two points yield the same label if and only if they are path-connected. To robustly handle various kinds of edge cases, the algorithm may assign identical labels to distinct components, but only when they are exactly or nearly touching, relative to a user-controlled length scale. A variety of numerical experiments assess the effectiveness of the overall approach, including statistical analyses on randomly generated multi-component geometry in 2D and 3D, as well as specific examples involving cusps, self-intersections, junctions, and other kinds of singularities.