Journal of Mathematical Imaging and Vision

, Volume 3, Issue 2, pp 205–221 | Cite as

On the flatness of digital hyperplanes

  • Peter Veelaert
Article

Abstract

This paper investigates the properties of digital hyperplanes of arbitrary dimension. We extend previous results that have been obtained for digital straight lines and digital planes, namely, Hung's evenness, Rosenfeld's chord, and Kim's chordal triangle property. To characterize digital hyperplanes we introduce the notion of digital flatness. We make a distinction between flatness and local flatness. The main tool we use is Helly's First Theorem, a classical result on convex sets, by means of which precise and verifiable conditions are given for the flatness of digital point sets. The main result is the proof of the equivalence of local flatness, evenness, and the chord property for certain infinite digital point sets in spaces of arbitrary dimension.

Key words

vision geometry digital geometry digitization scheme digital hyperplanes chord property evenness property 

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References

  1. 1.
    C. Ronse, “A bibliography on digital and computational convexity (1961–1988),”IEEE Trans. Patt. Anal. Mach. Intell., vol. 11, 1989, pp. 181–190.Google Scholar
  2. 2.
    S.H.Y. Hung, “On the straightness of digital arcs,”IEEE Trans. Patt. Anal. Mach. Intell., vol. 7, 1985, pp. 203–215.Google Scholar
  3. 3.
    A. Rosenfeld, “Digital straight line segments,”IEEE Trans. Comput., vol. 23, 1974, pp. 1264–1269.Google Scholar
  4. 4.
    C. Ronse, “A simple proof of Rosenfeld's characterization of digital straight line segments,”Patt. Recog. Lett., vol. 3, 1985, pp. 323–326.Google Scholar
  5. 5.
    C. Ronse, “Criteria for approximation of linear and affine functions,”Arch. Math., vol. 46, 1986, pp. 371–384.Google Scholar
  6. 6.
    C. Ronse, “A note on the approximation of linear and affine functions: the case of bounded slope,”Arch. Math., vol. 54, 1990, pp. 601–609.Google Scholar
  7. 7.
    C.E. Kim, “Three-dimensional digital planes,”IEEE Trans. Patt. Anal. Mach. Intell., vol. 6, 1984, pp. 639–645.Google Scholar
  8. 8.
    I. Stojmenović and R. Tošić, “Digitization schemes and the recognition of digital straight lines, hyperplanes, and flats in arbitrary dimensions,” inVision Geometry, Contemporary Mathematics Series Vol. 119, R.A. Melter, A. Rosenfeld, and P. Bhattacharya, eds., American Mathematical Society, Providence, RI, 1991, pp. 197–212.Google Scholar
  9. 9.
    J. Stoer and C. Witzgall,Convexity and Optimization in Finite Dimensions I. Springer, Berlin, 1970.Google Scholar
  10. 10.
    G.H. Hardy and E.M. Wright,An Introduction to the Theory of Numbers. Clarendon, Oxford, 1979.Google Scholar
  11. 11.
    P.J. Kelly and M.L. Weiss,Geometry and Convexity: A Study in Mathematical Methods. Wiley, New York, 1979.Google Scholar
  12. 12.
    P. Veelaert, “Digital planarity of rectangular surface segments,”IEEE Trans. Patt. Anal. Mach. Intell., to appear.Google Scholar

Copyright information

© Kluwer Academic Publishers 1993

Authors and Affiliations

  • Peter Veelaert
    • 1
  1. 1.Electronics LaboratoryUniversity of GhentGhentBelgium

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