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Weighted Distances and Digital Disks on the Khalimsky Grid

Disks with Holes and Islands

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Abstract

In this paper, after providing an appropriate coordinate system, we investigate the weighted distances on the Khalimsky grid. There are two types of natural neighborhood relations, and one semi-neighborhood is also defined. Weighted distances are defined for both cases, i.e., allowing or not the semi-neighborhood. We give formulae for computing the weighted distance of any point-pair on the Khalimsky grid in these cases. Digital disks based on the weighted distances are also investigated. In some cases, these disks may not be convex; moreover, they may contain holes. Sometimes, if semi-neighborhood is allowed, they are not connected, i.e., they contain islands. The conditions of concavities, holes and islands are characterized.

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Acknowledgements

Some parts of the paper were presented in the 19th IAPR international conference on Discrete Geometry for Computer Imagery (DGCI2016) Nantes, France, see [12]. Comments and remarks of the participants, comments and advices of the anonymous reviewers are gratefully acknowledged.

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Authors and Affiliations

  1. Edutus College, Tatabánya, Hungary

    Gergely Kovács

  2. Department of Mathematics, Faculty of Arts and Sciences, Eastern Mediterranean University, Mersin-10, Famagusta, North Cyprus, Turkey

    Benedek Nagy

  3. Department of Industrial Engineering, Eastern Mediterranean University, Mersin-10, Famagusta, North Cyprus, Turkey

    Béla Vizvári

Authors
  1. Gergely Kovács
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  2. Benedek Nagy
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  3. Béla Vizvári
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Correspondence to Benedek Nagy.

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Kovács, G., Nagy, B. & Vizvári, B. Weighted Distances and Digital Disks on the Khalimsky Grid. J Math Imaging Vis 59, 2–22 (2017). https://doi.org/10.1007/s10851-016-0701-5

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