Introduction to Digital Level Layers

  • Yan Gérard
  • Laurent Provot
  • Fabien Feschet
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6607)


We introduce the notion of Digital Level Layer, namely the subsets of \(\mathbb Z ^d\) characterized by double-inequalities \(h_1 \preccurlyeq f(x) \curlyeqprec h_2\). The purpose of the paper is first to investigate some theoretical properties of this class of digital primitives according to topological and morphological criteria. The second task is to show that even if we consider functions f of high degree, the computations on Digital Level Layers, for instance the computation of a DLL containing an input set of points, remain linear. It makes this notion suitable for applications, for instance to provide analytical characterizations of digital shapes.


Digital Primitives Cover Thickness Linear Programming Support Vector Machines 


  1. 1.
    Aizerman, M., Braverman, E., Rozonoer, L.: Theoretical foundations of the potential function method in pattern recognition learning. Automation and Remote Control 25, 821–837 (1964)zbMATHGoogle Scholar
  2. 2.
    Andres, E.: The supercover of an m-flat is a discrete analytical object. Theoretical Computer Science 406(1-2), 8–14 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Andres, E., Acharya, R., Sibata, C.: Discrete analytical hyperplanes. Graphical Model and Image Processing 59(5), 302–309 (1997)CrossRefGoogle Scholar
  4. 4.
    Andres, E., Jacob, M.A.: The discrete analytical hyperspheres. IEEE Transactions on Visualization and Computer Graphics 3(1), 75–86 (1997)CrossRefGoogle Scholar
  5. 5.
    Brimkov, V.E., Andres, E., Barneva, R.P.: Object discretizations in higher dimensions. Pattern Recognition Letters 23(6), 623–636 (2002)CrossRefzbMATHGoogle Scholar
  6. 6.
    Cohen-Or, D., Kaufman, A.E.: Fundamentals of surface voxelization. Graphical Model and Image Processing 57(6), 453–461 (1995)CrossRefGoogle Scholar
  7. 7.
    Cristianini, N., Shawe-Taylor, J.: Support Vector Machines and other kernel-based learning methods. Cambridge University Press, Cambridge (2000)CrossRefzbMATHGoogle Scholar
  8. 8.
    Domenjoud, E., Jamet, D., Toutant, J.L.: On the connecting thickness of arithmetical discrete planes. In: Brlek, S., Reutenauer, C., Provençal, X. (eds.) DGCI 2009. LNCS, vol. 5810, pp. 362–372. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  9. 9.
    Gérard, Y., Debled-Rennesson, I., Zimmermann, P.: An elementary digital plane recognition algorithm. Discrete Applied Mathematics 151, 169–183 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Gilbert, E.G., Johnson, D.W., Keerthi, S.S.: A fast procedure for computing the distance between complex objects in three-dimensional space. IEEE Journal of Robotics and Automation 4, 193–203 (1988)CrossRefGoogle Scholar
  11. 11.
    Jamet, D., Toutant, J.L.: Minimal arithmetic thickness connecting discrete planes. Discrete Applied Mathematics 157(3), 500–509 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Megiddo, N.: Linear programming in linear time when the dimension is fixed. Journal of the ACM 31(1), 114–127 (1984)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Reveillès, J.P.: Géométrie discrète, calculs en nombre entiers et algorithmique. Thèse d’état, Université Louis Pasteur, Strasbourg (1991)Google Scholar
  14. 14.
    Toussaint, G.T.: Solving geometric problems with the rotating calipers. In: Proceedings of IEEE MELECON 1983, Athens, Greece (1983)Google Scholar
  15. 15.
    Vapnik, V.: Statistical Learning Theory. Wiley, New York (1998)zbMATHGoogle Scholar
  16. 16.
    Ziegler, G.M.: Lectures on polytopes. Springer, Heidelberg (1995)CrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Yan Gérard
    • 1
  • Laurent Provot
    • 1
  • Fabien Feschet
    • 1
  1. 1.ISITUniv. Clermont 1AubièreFrance

Personalised recommendations