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Optimal conditions for connectedness of discretized sets

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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Fig. 1
Fig. 2

Notes

  1. 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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Acknowledgements

The authors thank the two anonymous referees for their useful remarks and suggestions. The results of Sect. 3 have been presented in the proceedings of the 20th International Workshop on Combinatorial Image Analysis (Brimkov and Brimkov 2020).

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Correspondence to Boris Brimkov.

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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

Keywords

  • Discrete geometry
  • Geometric features
  • Connected set
  • Discrete connectivity
  • Connectivity control
  • Offset discretization