Revisiting Digital Straight Segment Recognition

  • François de Vieilleville
  • Jacques-Olivier Lachaud
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4245)

Abstract

This paper presents new results about digital straight segments, their recognition and related properties. They come from the study of the arithmetically based recognition algorithm proposed by I. Debled-Rennesson and J.-P. Reveillès in 1995 [1]. We indeed exhibit the relations describing the possible changes in the parameters of the digital straight segment under investigation. This description is achieved by considering new parameters on digital segments: instead of their arithmetic description, we examine the parameters related to their combinatoric description. As a result we have a better understanding of their evolution during recognition and analytical formulas to compute them. We also show how this evolution can be projected onto the Stern-Brocot tree. These new relations have interesting consequences on the geometry of digital curves. We show how they can for instance be used to bound the slope difference between consecutive maximal segments.

References

  1. 1.
    Debled-Renesson, I., Reveillès, J.P.: A linear algorithm for segmentation of discrete curves. Int. J. Pattern Recognit. Artif. Intell. 9, 635–662 (1995)CrossRefGoogle Scholar
  2. 2.
    Klette, R., Rosenfeld, A.: Digital straightness: a review. Discrete Appl. Math. 139(1-3), 197–230 (2004)MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Dorst, L., Smeulders, A.W.M.: Discrete representation of straight lines. IEEE Trans. Pattern Anal. Mach. Intell. 6, 450–463 (1984)MATHCrossRefGoogle Scholar
  4. 4.
    Coeurjolly, D.: Supercover model and digital straight line recognition on irregular isothetic grids. In: Andrès, É., Damiand, G., Lienhardt, P. (eds.) DGCI 2005. LNCS, vol. 3429, pp. 311–322. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  5. 5.
    Berstel, J., De Luca, A.: Sturmian words, lyndon words and trees. Theoret. Comput. Sci. 178(1-2), 171–203 (1997)MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Voss, K.: Discrete Images, Objects, and Functions in ℤn. Springer, Heidelberg (1993)Google Scholar
  7. 7.
    de Vieilleville, F., Lachaud, J.-O., Feschet, F.: Maximal digital straight segments and convergence of discrete geometric estimators. In: Kalviainen, H., Parkkinen, J., Kaarna, A. (eds.) SCIA 2005. LNCS, vol. 3540, pp. 988–997. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  8. 8.
    de Vieilleville, F., Lachaud, J.O., Feschet, F.: Maximal digital straight segments and convergence of discrete geometric estimators. Research Report 1350-05, LaBRI, University Bordeaux 1, Talence, France (2005)Google Scholar
  9. 9.
    Feschet, F.: Optimal time computation of the tangent of a discrete curve: Application to the curvature. In: Bertrand, G., Couprie, M., Perroton, L. (eds.) DGCI 1999. LNCS, vol. 1568, pp. 31–40. Springer, Heidelberg (1999)CrossRefGoogle Scholar
  10. 10.
    Klette, R., Rosenfeld, A.: Digital Geometry - Geometric Methods for Digital Picture Analysis. Morgan Kaufmann, San Francisco (2004)MATHGoogle Scholar
  11. 11.
    Debled-Rennesson, I.: Etude et reconnaissance des droites et plans discrets. PhD thesis, Université Louis Pasteur, Strasbourg, France (1995)Google Scholar
  12. 12.
    Feschet, F., Tougne, L.: On the min DSS problem of closed discrete curves. In: Del Lungo, A., Di Gesù, V., Kuba, A. (eds.) IWCIA. Electonic Notes in Discrete Math., vol. 12, Elsevier, Amsterdam (2003)Google Scholar
  13. 13.
    Lachaud, J.-O., Vialard, A., de Vieilleville, F.: Analysis and comparative evaluation of discrete tangent estimators. In: Andrès, É., Damiand, G., Lienhardt, P. (eds.) DGCI 2005. LNCS, vol. 3429, pp. 240–251. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  14. 14.
    Reiter-Doerksen, H., Debled-Rennesson, I.: Convex and concave parts of digital curves. In: Dagstuhl Seminar, Geometric Properties from Incomplete Data (2004)Google Scholar
  15. 15.
    Reveillès, J.P.: Géométrie discrète, calcul en nombres entiers et algorithmique. Thèse d’etat, Université Louis Pasteur, Strasbourg (1991)Google Scholar
  16. 16.
    Sivignon, I., Dupont, F., Chassery, J.-M.: New results about digital intersections. In: Nyström, I., Sanniti di Baja, G., Svensson, S. (eds.) DGCI 2003. LNCS, vol. 2886, pp. 102–113. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  17. 17.
    Sivignon, I.: De la caractérisation des primitives à la reconstruction polyédrique de surfaces en géométrie discrète. PhD thesis, Institut national polytechnique de Grenoble, France (2004)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • François de Vieilleville
    • 1
  • Jacques-Olivier Lachaud
    • 1
  1. 1.LaBRIUniv. Bordeaux 1TalenceFrance

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