The Offset Filtration of Convex Objects

TL;DR

This paper introduces a method to compute filtrations of offsets of convex objects using Voronoi diagrams, providing efficient algorithms and demonstrating improved topological stability over point sampling methods.

Contribution

It presents a novel approach to generate filtrations for convex object offsets based on Voronoi diagrams, with proven efficiency and stability in 2D and 3D.

Findings

Filtration size proportional to Voronoi diagram size in 2D and 3D.

Algorithm runs in $ heta(n \log n)$ for 2D and expected $O(n^{3+\epsilon})$ for 3D.

Experiments show improved topological stability over point sampling methods.

Abstract

We consider offsets of a union of convex objects. We aim for a filtration, a sequence of nested cell complexes, that captures the topological evolution of the offsets for increasing radii. We describe methods to compute a filtration based on the Voronoi partition with respect to the given convex objects. We prove that, in two and three dimensions, the size of the filtration is proportional to the size of the Voronoi diagram. Our algorithm runs in $\Theta(n \log{n})$ in the $2$-dimensional case and in expected time $O(n^{3 + \epsilon})$, for any $\epsilon > 0$, in the $3$-dimensional case. Our approach is inspired by alpha-complexes for point sets, but requires more involved machinery and analysis primarily since Voronoi regions of general convex objects do not form a good cover. We show by experiments that our approach results in a similarly fast and topologically more stable method for computing a filtration compared to approximating the input by point samples.