Discrete & Computational Geometry

, Volume 42, Issue 3, pp 359–378 | Cite as

Consistent Digital Rays

  • Jinhee Chun
  • Matias Korman
  • Martin Nöllenburg
  • Takeshi Tokuyama
Article

Abstract

Given a fixed origin o in the d-dimensional grid, we give a novel definition of digital rays dig(op) from o to each grid point p. Each digital ray dig(op) approximates the Euclidean line segment \(\overline {op}\) between o and p. The set of all digital rays satisfies a set of axioms analogous to the Euclidean axioms. We measure the approximation quality by the maximum Hausdorff distance between a digital ray and its Euclidean counterpart and establish an asymptotically tight Θ(log n) bound in the n×n grid. The proof of the bound is based on discrepancy theory and a simple construction algorithm. Without a monotonicity property for digital rays the bound is improved to O(1). Digital rays enable us to define the family of digital star-shaped regions centered at o, which we use to design efficient algorithms for image processing problems.

Keywords

Digital geometry Discrete geometry Star-shaped regions Tree embedding 

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Copyright information

© Springer Science+Business Media, LLC 2009

Authors and Affiliations

  • Jinhee Chun
    • 1
  • Matias Korman
    • 1
  • Martin Nöllenburg
    • 2
  • Takeshi Tokuyama
    • 1
  1. 1.Graduate School of Information SciencesTohoku UniversitySendaiJapan
  2. 2.Faculty of InformaticsKarlsruhe University and Karlsruhe Institute of Technology (KIT)KarlsruheGermany

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