Digital Distances and Integer Sequences

  • Nicolas Normand
  • Robin Strand
  • Pierre Evenou
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7749)


In recent years, the theory behind distance functions defined by neighbourhood sequences has been developed in the digital geometry community. A neighbourhood sequence is a sequence of integers, where each element defines a neighbourhood. In this paper, we establish the equivalence between the representation of convex digital disks as an intersection of half-planes (\(\mathcal{H}\)-representation) and the expression of the distance as a maximum of non-decreasing functions.

Both forms can be deduced one from the other by taking advantage of the Lambek-Moser inverse of integer sequences.

Examples with finite sequences, cumulative sequences of periodic sequences and (almost) Beatty sequences are given. In each case, closed-form expressions are given for the distance function and \(\mathcal{H}\)-representation of disks. The results can be used to compute the pair-wise distance between points in constant time and to find optimal parameters for neighbourhood sequences.


Distance Function Convex Polyhedron Integer Sequence Neighbourhood Sequence Discrete Distance 
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Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Nicolas Normand
    • 1
  • Robin Strand
    • 2
  • Pierre Evenou
    • 1
  1. 1.IRCCyN UMR CNRS 6597, Polytech NantesLUNAM Université, Université de NantesNantes Cedex 3France
  2. 2.Centre for Image AnalysisUppsala UniversitySweden

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