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A Relational Generalization of the Khalimsky Topology

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Combinatorial Image Analysis (IWCIA 2017)

Part of the book series: Lecture Notes in Computer Science ((LNIP,volume 10256))

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Abstract

We discuss certain n-ary relations (\(n>1\) an integer) and show that each of them induces a connectedness on its underlying set. Of these n-ary relations, we study a particular one on the digital plane \(\mathbb Z^2\) for every integer \(n>1\). As the main result, for each of the n-ary relations studied, we prove a digital analogue of the Jordan curve theorem for the induced connectedness. It follows that these n-ary relations may be used as convenient structures on the digital plane for the study of geometric properties of digital images. For \(n=2\), such a structure coincides with the (specialization order of the) Khalimsky topology and, for \(n>2\), it allows for a variety of Jordan curves richer than that provided by the Khalimsky topology.

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Acknowledgement

This work was supported by the Brno University of Technology Specific Research Program, project no. FSI-S-17-4464.

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Authors and Affiliations

  1. Institute of Mathematics, Brno University of Technology, Brno, Czech Republic

    Josef Šlapal

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  1. Josef Šlapal
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Correspondence to Josef Šlapal .

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Editors and Affiliations

  1. SUNY Buffalo State, Buffalo, New York, USA

    Valentin E. Brimkov

  2. Department of Applied Professional Studies, SUNY Fredonia, Fredonia, New York, USA

    Reneta P. Barneva

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Šlapal, J. (2017). A Relational Generalization of the Khalimsky Topology. In: Brimkov, V., Barneva, R. (eds) Combinatorial Image Analysis. IWCIA 2017. Lecture Notes in Computer Science(), vol 10256. Springer, Cham. https://doi.org/10.1007/978-3-319-59108-7_11

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  • DOI: https://doi.org/10.1007/978-3-319-59108-7_11

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-59107-0

  • Online ISBN: 978-3-319-59108-7

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