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Optimal conditions for connectedness of discretized sets

Journal of Combinatorial Optimization volume 43, pages 1493–1506 (2022)Cite this article

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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Authors and Affiliations

  1. Department of Mathematics and Statistics, Slippery Rock University, Slippery Rock, PA, 16057, USA

    Boris Brimkov

  2. Mathematics Department, SUNY Buffalo State, Buffalo, NY, 14222, USA

    Valentin E. Brimkov

  3. Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, Sofia, 1113, Bulgaria

    Valentin E. Brimkov

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  1. Boris Brimkov
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  2. Valentin E. Brimkov
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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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Keywords

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