Digital Convexity and Cavity Trees

  • Gisela Klette
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8334)


The notion convexity has a long history in mathematics. It is a useful concept to describe shapes, functions, smoothness of curves or boundaries, and it has applications in many fields. Researchers apply different definitions for digital convexity to adapt known concepts from the continuous space, and to make use of proven theories and results. We review different approaches and we propose cavity trees for (a) analyzing the convexity of digital objects, (b) to decompose those objects into meaningful parts, and (c) to show an easy way to find convex and concave parts of a boundary of a digital region.


shape analysis feature extraction minimum-perimeter polygon minimum-length polygon cavity tree digital convexity geometric estimators 


  1. 1.
    Brlek, S., Lachaud, J.-O., Provençal, X.: Combinatorial view of digital convexity. In: Coeurjolly, D., Sivignon, I., Tougne, L., Dupont, F. (eds.) DGCI 2008. LNCS, vol. 4992, pp. 57–68. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  2. 2.
    Debled-Rennesson, I., Remy, J.-L., Rouyer-Degli, J.: Detection of discrete convexity of polyominoes. Discrete Applied Mathematics 125, 115–133 (2003)CrossRefMATHMathSciNetGoogle Scholar
  3. 3.
    Doerksen-Reiter, H., Debled-Rennesson, I.: Convex and concave parts of digital curves. Computational Imaging and Vision 31, 145–160 (2006)Google Scholar
  4. 4.
    Eckhardt, U., Doerksen-Reiter, H.: Polygonal representations of digital sets. Algorithmica 38(1), 5–23 (2004)CrossRefMATHMathSciNetGoogle Scholar
  5. 5.
    Kiselman, C.O.: Characterizing digital straightness and digital convexity by means of difference operators. Mathematika 57, 355–380 (2011)CrossRefMATHMathSciNetGoogle Scholar
  6. 6.
    Klette, R., Zunic, J.: Multigrid convergence of calculated features in image analysis. J. Mathematical Imaging Vision 13, 173–191 (2000)CrossRefMATHMathSciNetGoogle Scholar
  7. 7.
    Klette, R., Rosenfeld, A.: Digital Geometry – Geometric Methods for Digital Picture Analysis. Morgan Kaufmann, San Francisco (2004)MATHGoogle Scholar
  8. 8.
    Klette, G.: Recursive calculation of relative convex hulls. In: Debled-Rennesson, I., Domenjoud, E., Kerautret, B., Even, P. (eds.) DGCI 2011. LNCS, vol. 6607, pp. 260–271. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  9. 9.
    Klette, G.: Recursive Computation of Minimum-Length Polygons. Computer Vision and Image Understanding 117, 386–392 (2012)CrossRefGoogle Scholar
  10. 10.
    Kim, C.E.: Digital convexity, straightness, and convex polygons. PAMI 4, 618–626 (1982)CrossRefMATHGoogle Scholar
  11. 11.
    Kim, C.E., Rosenfeld, A.: Digital straight lines and convexity of digital regions. PAMI 4, 149–153 (1982)CrossRefMATHGoogle Scholar
  12. 12.
    Lachaud, J.O., Vialard, A., de Vieilleville, F.: Fast, accurate and convergent tangent estimation on digital contours. Image and Vision Computing 25, 1572–1587 (2007)CrossRefGoogle Scholar
  13. 13.
    Li, F., Klette, R.: Euclidean Shortest Paths. Springer, London (2011)CrossRefMATHGoogle Scholar
  14. 14.
    Melkman, A.: On-line construction of the convex hull of a simple polygon. Information Processing Letters 25, 11–12 (1987)CrossRefMATHMathSciNetGoogle Scholar
  15. 15.
    Minsky, M., Papert, S.: Perceptrons. MIT Press, Reading (1969)MATHGoogle Scholar
  16. 16.
    Provençal, X., Lachaud, J.-O.: Two linear-time algorithms for computing the minimum length polygon of a digital contour. In: Brlek, S., Reutenauer, C., Provençal, X. (eds.) DGCI 2009. LNCS, vol. 5810, pp. 104–117. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  17. 17.
    Papadopoulus, A.: Metric Spaces, Convexity and Nonpositive Curvature. European Mathematical Society (2005)Google Scholar
  18. 18.
    Roussillon, T., Tougne, L., Sivignon, I.: What does digital straightness tell about digital convexity? In: Wiederhold, P., Barneva, R.P. (eds.) IWCIA 2009. LNCS, vol. 5852, pp. 43–55. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  19. 19.
    Roussillon, T., Sivignon, I.: Reversible polygon that faithfully represents the convex and concave parts of a digital curve. Pattern Recognition 44, 2693–2700 (2011)CrossRefMATHGoogle Scholar
  20. 20.
    Rosenfeld, A.: Picture Processing by Computer. Academic Press, New York (1969)MATHGoogle Scholar
  21. 21.
    Sklansky, J.: Recognition of convex blobs. Pattern Recognition 2, 3–10 (1970)CrossRefGoogle Scholar
  22. 22.
    Sklansky, J.: Measuring cavity on a rectangular mosaic. IEEE Trans. Computing 21, 1355–1364 (1972)CrossRefMATHMathSciNetGoogle Scholar
  23. 23.
    Sloboda, F., Stoer, J.: On piecewise linear approximation of planar Jordan curves. J. Computational and Applied Mathematics 55, 369–383 (1994)CrossRefMATHMathSciNetGoogle Scholar
  24. 24.
    Sloboda, F., Zatko, B., Stoer, J.: On approximation of planar one dimensional continua. In: Klette, R., Rosenfeld, A., Sloboda, F. (eds.) Advances in Digital and Computational Geometry, pp. 113–160 (1998)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  • Gisela Klette
    • 1
  1. 1.AUT UniversityAucklandNew Zealand

Personalised recommendations