Convexity Preserving Contraction of Digital Sets

  • Lama TarsissiEmail author
  • David Coeurjolly
  • Yukiko Kenmochi
  • Pascal Romon
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 12047)


Convexity is one of the useful geometric properties of digital sets in digital image processing. There are various applications which require deforming digital convex sets while preserving their convexity. In this article, we consider the contraction of such digital sets by removing digital points one by one. For this aim, we use some tools of combinatorics on words to detect a set of removable points and to define such convexity-preserving contraction of a digital set as an operation of re-writing its boundary word. In order to chose one of removable points for each contraction step, we present three geometrical strategies, which are related to vertex angle and area changes. We also show experimental results of applying the methods to repair some non-convex digital sets, which are obtained by rotations of convex digital sets.


Digital convexity Digital set contraction Christoffel words Lyndon words 


  1. 1.
    Berstel, J., Lauve, A., Reutenauer, C., Saliola, F.: Combinatorics on words: Christoffel words and repetition in words (2008)Google Scholar
  2. 2.
    Borel, J.-P., Laubie, F.: Quelques mots sur la droite projective réelle. J. de théorie des nombres de Bordeaux 5(1), 23–51 (1993)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Brlek, S., Lachaud, J.-O., Provençal, X., Reutenauer, C.: Lyndon+ Christoffel=digitally convex. Pattern Recognit. 42(10), 2239–2246 (2009) CrossRefGoogle Scholar
  4. 4.
    Castiglione, G., Frosini, A., Munarini, E., Restivo, A., Rinaldi, S.: Combinatorial aspects of L-convex polyominoes. Eur. J. Comb. 28(6), 1724–1741 (2007)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Castiglione, G., Frosini, A., Restivo, A., Rinaldi, S.: Enumeration of l-convex polyominoes by rows and columns. Theor. Comput. Sci. 347(1–2), 336–352 (2005)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Charrier, E., Buzer, L.: Approximating a real number by a rational number with a limited denominator: a geometric approach. Discrete Appl. Math. 157(16), 3473–3484 (2009)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Chen, K., Fox, R., Lyndon, R.: Free differential calculus IV. The quotient groups of the lower central series. Ann. Math. 68, 81–95 (1958) MathSciNetCrossRefGoogle Scholar
  8. 8.
    Christoffel, E.: Observatio arithmetica. Annali di Matematica Pura ed Applicata (1867–1897), 6(1), 148–152 (1875)CrossRefGoogle Scholar
  9. 9.
    Del Lungo, A., Duchi, E., Frosini, A., Rinaldi, S.: Enumeration of convex polyominoes using the ECO method. In: DMCS, pp. 103–116 (2003)Google Scholar
  10. 10.
    Del Lungo, A., Duchi, E., Frosini, A., Rinaldi, S.: On the generation and enumeration of some classes of convex polyominoes. Electron. J. Comb. 11(1), 60 (2004)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Dulio, P., Frosini, A., Rinaldi, S., Tarsissi, L., Vuillon, L.: First steps in the algorithmic reconstruction of digital convex sets. In: Brlek, S., Dolce, F., Reutenauer, C., Vandomme, É. (eds.) WORDS 2017. LNCS, vol. 10432, pp. 164–176. Springer, Cham (2017). Scholar
  12. 12.
    Duval, J.: Mots de lyndon et périodicité. RAIRO, Informatique théorique 14(2), 181–191 (1980)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Freeman, H.: On the encoding of arbitrary geometric configurations. IRE Trans. Electron. Comput. 2, 260–268 (1961)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Hayes, A.C., Larman, D.G.: The vertices of the knapsack polytope. Discrete Appl. Math. 6(2), 135–138 (1983)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Klette, R., Rosenfeld, A.: Digital Geometry: Geometric Methods for Digital Picture Analysis. Morgan Kaufmann Publishers Inc., San Francisco (2004)zbMATHGoogle Scholar
  16. 16.
    Lothaire, M.: Algebraic Combinatorics on Words. Encyclopedia of Mathematics and Its Applications, vol. 90. Cambridge University Press, Cambridge (2002)CrossRefGoogle Scholar
  17. 17.
    Lyndon, R.: Identities in finite algebras. Proc. Am. Math. Soc. 5(1), 8–9 (1954)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Minsky, M., Papert, S.: Perceptrons. MIT Press, Cambridge (1969) zbMATHGoogle Scholar
  19. 19.
    Pick, G.: Geometrisches zur zahlenlehre. Sitzungsberichte des Deutschen Naturwissenschaftlich-Medicinischen Vereines für Böhmen “Lotos” in Prag., vol. 47–48, pp. 1899–1900 (1906)Google Scholar

Copyright information

© Springer Nature Switzerland AG 2020

Authors and Affiliations

  • Lama Tarsissi
    • 1
    • 2
    Email author
  • David Coeurjolly
    • 3
  • Yukiko Kenmochi
    • 1
  • Pascal Romon
    • 2
  1. 1.LIGM, Université Gustave Eiffel, CNRS, ESIEE ParisMarne-la-valléeFrance
  2. 2.LAMA, Université Gustave Eiffel, CNRSMarne-la-valléeFrance
  3. 3.Université de Lyon, CNRS, LIRISLyonFrance

Personalised recommendations