Abstract
An offset discretization of a set \(X \subset {\mathbb {R}}^n\) is obtained by taking the integer points inside a closed neighborhood of X of a certain radius. In this work we determine a minimum threshold for the offset radius, beyond which the discretization of an arbitrary (possibly disconnected) set is always connected. The obtained results hold for a broad class of disconnected subsets of \({\mathbb {R}}^n\) and generalize several previous results. We also extend our results to infinite discretizations of unbounded subsets of \({\mathbb {R}}^n\) and consider certain algorithmic aspects. The obtained results can be applied to component topology preservation as well as extracting geometric features and connectivity control of very large object discretizations.
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Notes
The function g itself, defined on the subsets of \({\mathbb {R}}^n\) is called a gap functional. See, e.g. Beer (1993) for more details.
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Brimkov, B., Brimkov, V.E. Optimal conditions for connectedness of discretized sets. J Comb Optim 43, 1493–1506 (2022). https://doi.org/10.1007/s10878-020-00691-0
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DOI: https://doi.org/10.1007/s10878-020-00691-0