The Offset Filtration of Convex Objects

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 Θ(n log n) in the 2-dimensional case and in expected time O(n 3+ε ), for any ε > 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.

Offset hypersurfaces and persistent homology of algebraic varieties

“…In what follows, we study the persistent homology of the offset filtration of an algebraic variety, which we define to be the homology of its offsets. Related work includes [14] in which the notion of persistent homology is extended to the offsets of convex objects.…”