Digital Distances and Integer Sequences

Nicolas Normand (1) , Robin Strand (2) , Pierre Evenou (1)

1 IRCCyN - Institut de Recherche en Communications et en Cybernétique de Nantes
2 CBA - Centre for Image Analysis

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 (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 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.

Domaines: Traitement du signal et de l'image [eess.SP]

Nicolas Normand, Robin Strand, Pierre Evenou. Digital Distances and Integer Sequences. 17th IAPR International Conference, DGCI, Mar 2013, Séville, Spain. pp.169-179, ⟨10.1007/978-3-642-37067-0_15⟩. ⟨hal-01155038⟩