Optimal Covering of a Straight Line Applied to Discrete Convexity

  • Jean-Marc Chassery
  • Isabelle Sivignon
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7749)


The relation between a straight line and its digitization as a digital straight line is often expressed using a notion of proximity. In this contribution, we consider the covering of the straight line by a set of balls centered on the digital straight line pixels. We prove that the optimal radius of the balls is strictly less than one, and can be expressed as a function of the slope of the straight line. This property is used to define discrete convexity in concordance with previous works on convexity.


Convex Hull Convex Shape Optimal Covering Digital Space Optimal Radius 
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  1. 1.
    Chassery, J.M.: Discrete convexity: Definition, parametrization, and compatibility with continuous convexity. Computer Vision, Graphics, and Image Processing 21(3), 326–344 (1983)zbMATHCrossRefGoogle Scholar
  2. 2.
    Coeurjolly, D., Montanvert, A., Chassery, J.M.: Géométrie discrète et images numériques. Hermès, traité IC2, série signal et image (2007)Google Scholar
  3. 3.
    Debled-Rennesson, I., Reveillès, J.P.: A linear algorithm for segmentation of digital curves. International Journal of Pattern Recognition and Artificial Intelligence 9(6), 635–662 (1995)CrossRefGoogle Scholar
  4. 4.
    Eckhardt, U.: Digital Lines and Digital Convexity. In: Bertrand, G., Imiya, A., Klette, R. (eds.) Digital and Image Geometry. LNCS, vol. 2243, pp. 209–228. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  5. 5.
    Kim, C.E.: Digital convexity, straightness, and convex polygons 4(6), 618–626 (1982)Google Scholar
  6. 6.
    Kim, C.E., Rosenfeld, A.: Digital straight lines and convexity of digital regions 4(2), 149–153 (1982)Google Scholar
  7. 7.
    Kim, C.E., Sklansky, J.: Digital and cellular convexity. Pattern Recognition 15(5), 359–367 (1982)MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Klette, R., Rosenfeld, A.: Digital Geometry: Geometric methods for digital picture analysis. Morgan Kaufmann (2004)Google Scholar
  9. 9.
    Reveilles, J.P.: Géométrie discrète, calcul en nombres entiers et algorithmique. Thèse d’etat, Université Louis Pasteur, Strasbourg, France (1991)Google Scholar
  10. 10.
    Ronse, C.: A bibliography on digital and computational convexity (1961-1988) 11(2), 181–190 (1989)Google Scholar
  11. 11.
    Roussillon, T.: Algorithmes d’extraction de modèles géométriques discrets pour la représentation robuste des formes. Ph.D. thesis, Université Lumière Lyon 2 (2009)Google Scholar
  12. 12.
    Sklansky, J.: Recognition of convex blobs. Pattern Recognition 2(1), 3–10 (1970)CrossRefGoogle Scholar
  13. 13.
    Sklansky, J.: Measuring concavity on a rectangular mosaic. IEEE Transactions on Computers (12), 1355–1364 (1972)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Jean-Marc Chassery
    • 1
  • Isabelle Sivignon
    • 1
  1. 1.CNRS, UMR 5216gipsa-labFrance

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