High dimensional consistent digital segments

Man Kwun Chiu, Matias Korman

Research output: Contribution to journalArticlepeer-review

1 Citation (Scopus)

Abstract

We consider the problem of digitalizing Euclidean line segments from ℝ d to ℤ d . Christ, Pálvölgyi, and Stojaković, [Discrete Comput. Geom., 47(2012), pp. 691-710] showed how to construct a set of consistent digital segments (CDS) for d = 2: a collection of segments connecting any two points in ℤ 2 that satisfies the natural extension of the Euclidean axioms to ℤ d . In this paper we study the construction of CDSs in higher dimensions. We extend some of their results to higher dimensions. Specifically we show that any total order can be used to create a set of consistent digital rays CDR in ℤ d (a set of rays emanating from a fixed point p that satisfies the extension of the Euclidean axioms). Then we use the same approach to create a CDS in ℤ d and observe that it only works in some cases. We fully characterize for which total orders the construction is consistent (and thus gives a CDS). In particular, this positively answers the question posed by Christ and co-authors.

Original languageEnglish
Pages (from-to)2566-2590
Number of pages25
JournalSIAM Journal on Discrete Mathematics
Volume32
Issue number4
DOIs
Publication statusPublished - 2018

Keywords

  • Computer vision
  • Consistent digital line segments
  • Digital geometry
  • Image segmentation
  • Integer lattice

ASJC Scopus subject areas

  • Mathematics(all)

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