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 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 Θ(n logn) 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 compared to approximating the input by point samples.
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Work by D.H. and D.S. has been supported in part by the Israel Science Foundation (grant no. 1102/11), by the German-Israeli Foundation (grant no. 1150-82.6/2011), and by the Hermann Minkowski–Minerva Center for Geometry at Tel Aviv University. M.K. acknowledges support by the Max Planck Center of Visual Computing and Communication.
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Halperin, D., Kerber, M., Shaharabani, D. (2015). The Offset Filtration of Convex Objects. In: Bansal, N., Finocchi, I. (eds) Algorithms - ESA 2015. Lecture Notes in Computer Science(), vol 9294. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-48350-3_59
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