Recognizing arithmetic straight lines and planes

  • Jean Françon
  • Jean-Maurice Schramm
  • Mohamed Tajine
Conference paper
Part of the Lecture Notes in Computer Science book series


The problem of recognizing a straight line in the discrete plane ℤ2 (resp. a plane in ℤ3) is to find an algorithm deciding wether a given set of points in ℤ2 (resp. ℤ3) belongs to a line (resp. a plane). In this paper the lines and planes are arithmetic, as defined by Reveilles [Rev91], and the problem is translated, for any width that is a linear function of the coefficients of the normal to the searched line or plane, into the problem of solving a set of linear inequalities. This new problem is solved by using the Fourier's elimination algorithm. If there is a solution, the family of solutions is given by the algorithm as a conjunction of linear inequalities. This method of recognition is well suited to computer imagery, because any traversal algorithm of the given set is possible, and also because any incomplete segment of line or plane can be recognized.

Key words

Discrete plane discrete line recognition algorithm Fourier's algorithm 


  1. [Bor-Fra94]
    Ph. Borianne et J. Françon. Polyhédrisation réversible de volumes discrets. 4thConference on Discrete Geometry in Computer Imagery, Grenoble, 1994.Google Scholar
  2. [Dan63]
    G. B. Dantzig. Linear programming and extentions. Princeton, 1963.Google Scholar
  3. [Deb95]
    I. Debled-Renesson. Etude et reconnaissance des droites et plans discrets. Thèse de doctorat soutenue à l'Université Louis Pasteur de Strasbourg, 1995.Google Scholar
  4. [Deb-Rev94a]
    J.P. Reveilles, I. Debled-Renesson. A linear algorithm for segmentation of discrete curves. Third International Workshop on Parallel Image Analysis: Theory and Applications, Washington, 1994.Google Scholar
  5. [Deb-Rev94b]
    I. Debled-Renesson, J.P. Reveilles. An incremental algorithm for digital plan recognition. 4th Conference on Discrete Geometry in Computer Imagery, Strasbourg, 1994.Google Scholar
  6. [Deb-Rev94c]
    I. Debled, J.P. Reveillès. A new approach to digital planes. Vision Geometry III, Boston, 1994.Google Scholar
  7. [Fan56]
    K. Fan. On system of linear inequalities. In Kuhn-Tucker (eds.), Linear inequalities and related systems, Ann. of Math. Study 38: Princeton Univ. Press, 1956, p. 99–156.Google Scholar
  8. [Fou 1826]
    J. B. J. Fourier. Solution d'une question particulière du calcul des inégalités. Oeuvre II, Paris, 1826, p. 317–328.Google Scholar
  9. [Fra95]
    J. Françon. Arithmetic planes and combinatorial manifolds. 5th Conference on Discrete Geometry in Computer Imagery, Clermont-Ferrand, 1995.Google Scholar
  10. [Fra96]
    J. Françon. Sur la topologie d'un plan arithmétique. Theor. Comput. Sc. 156, 1996, p. 159–176.Google Scholar
  11. [Kov90]
    V.A. Kovalesky. New definition and fast recognition of digital straight line segments and arcs. Proc. of the 10th Intern. Conf. on Pattern Recognition, Atlantic City, IEEE Press, vol. II, 1990, p. 31–34.Google Scholar
  12. [Kro-Toc89]
    W.G. Kropatsch, H. Tockner. Detecting the straightness of digital curves in O(N) steps. Computer Vision, Graphics, and Image Processing 45, 1989, p. 1–21.Google Scholar
  13. [Kuh56]
    H. W. Kuhn. Solvability and consistency for linear equations and inequalities. Amer. Math. Monthly, Vol 63, 1956, p. 217–232.Google Scholar
  14. [Mat70]
    Yu. Matiiassevitch. Enumerable sets are diophantine. Doklady Akad. Nauk SSSR, 191, 1970, p. 279–282. (English translation: Soviet Math. Doklady, 1970, p. 354–357.Google Scholar
  15. [Mot36]
    T. S. Motzkin. Beitrage zur theorie der linearen Ungleichungen. Azriel, Jerusalem, 1936.Google Scholar
  16. [Pre29]
    M. Presburger. Üeber die vollständigkeit eines gewissen systems der arithmetik ganzen Zahlen in welchen die addition als einziger aperation hervortritt. Comptes Rendus du 1er Congrès des Math. des Pays Slaves, Warsaw, 1929, p. 92–101.Google Scholar
  17. [Rev91]
    J.P. Reveillès. Géométrie discrète, calcul en nombres entiers et algorithmique. Thèse d'état soutenue à l'Université Louis Pasteur, 1991.Google Scholar
  18. [Rev95]
    J.P. Reveilles. Combinatorial pieces in digital lines and planes. Vision Geometry 4, SPIE'95, San Diego, 1995.Google Scholar
  19. [Sto-Tos91]
    I. Stojmenovic, R. Tosic. Digitization schemes and the recognition of digital straight lines, hyperplanes, and flats in arbitrary dimensions. Contemporary Math., Vol. 119, 1991, p. 197–212.Google Scholar
  20. [Tar51]
    A. Tarski. A decision method for elementary algebra and geometry. Tech. Rep., University of California Press, Berkeley and Los Angeles, 1951.Google Scholar
  21. [Ver94]
    P. Veelaert. Digital planarity of rectangular planar segments. IEEE Trans. on P.A.M.I., 16, 1994, p. 647–652.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1996

Authors and Affiliations

  • Jean Françon
    • 1
  • Jean-Maurice Schramm
    • 1
  • Mohamed Tajine
    • 1
  1. 1.D'Informatique et de Télédétection Dépt. InformatiqueLaboratoire des Sciences de l'ImageStrasbourg cedex

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