TY - GEN

T1 - The offset filtration of convex objects

AU - Halperin, Dan

AU - Kerber, Michael

AU - Shaharabani, Doron

N1 - Publisher Copyright:
© Springer-Verlag Berlin Heidelberg 2015.

PY - 2015

Y1 - 2015

N2 - 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 diagram of 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 Ɵ(n log n) in the 2-dimensional case and in expected time O(n3+ 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 compared to approximating the input by point samples.

AB - 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 diagram of 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 Ɵ(n log n) in the 2-dimensional case and in expected time O(n3+ 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 compared to approximating the input by point samples.

UR - http://www.scopus.com/inward/record.url?scp=84945565023&partnerID=8YFLogxK

U2 - 10.1007/978-3-662-48350-3_59

DO - 10.1007/978-3-662-48350-3_59

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AN - SCOPUS:84945565023

SN - 9783662483497

T3 - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

SP - 705

EP - 716

BT - Algorithms – ESA 2015 - 23rd Annual European Symposium, Proceedings

A2 - Bansal, Nikhil

A2 - Finocchi, Irene

PB - Springer Verlag

Y2 - 14 September 2015 through 16 September 2015

ER -