(δ,ε)-ball approximation of a shape: definition and complexity

Dominique Attali 1 Tuong-Bach Nguyen 1 Isabelle Sivignon 1, *
* Auteur correspondant
1 GIPSA-AGPIG - AGPIG
GIPSA-DIS - Département Images et Signal
Abstract : Given a set S in Rn, a (δ,ε)-ball approximation of S is defined as a collection of balls that covers the morphological erosion of S (by a ball of radius ε) and remains inside the morphological dilation of S (by a ball of radius δ). We study the problem of computing a (δ,ε)-ball approximation when S is itself defined as a finite union of balls. This problem relates to geometric set cover problems but is however different in nature. It offers a new framework for reducing the size of a collection of balls while controlling both the inner and outer distance to the shape. We prove that computing a (δ,ε)-ball approximation of minimum cardinality is NP-complete for n = 2. Along the way, we study the boundary of unions of disks and their erosion, for which we derive a computational description.
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Journal of Discrete and Computational Geometry, Springer, In press, 〈10.1007/s00454-018-0019-8〉
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Dominique Attali, Tuong-Bach Nguyen, Isabelle Sivignon. (δ,ε)-ball approximation of a shape: definition and complexity. Journal of Discrete and Computational Geometry, Springer, In press, 〈10.1007/s00454-018-0019-8〉. 〈hal-01798844〉

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