A Composite and Quasi Linear Time Method for Digital Plane Recognition

  • Lilian Buzer
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4245)


This paper introduces a new method for the naive digital plane recognition problem. As efficient as existing alternatives, it is the only method known to the author that also guarantees a quasi linear time complexity in the worst case. The approach presented can be used to determine if a set of n points is a naive digital hyperplane in ℤ d in O(n log2 D) worst case time where D represents the size of a bounding box that encloses the points. In addition, the approach succeeds in reducing the naive digital plane recognition problem to a two-dimensional convex optimization program. Thus, the solution space is planar and only simple two-dimensional geometrical methods need to be applied during the recognition process. The algorithm is a composite of simple techniques based on one-dimensional optimization: Megiddo Oracle for linear programming and two-dimensional discrete geometry.


Normal Vector Side Length Search Domain Linear Programming Technique Binary Optimization 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


  1. 1.
    Brimkov, V., Coeurjolly, D., Klette, R.: Digital Planarity - A Review. Discrete Applied Mathematics (accepted for publication, 2006)Google Scholar
  2. 2.
    Buzer, L.: A linear incremental algorithm for naive and standard digital lines and planes recognition. Graphical Models 65(1-3), 61–76 (2003)MATHCrossRefGoogle Scholar
  3. 3.
    Megiddo, N.: Linear programming in linear time when the dimension is fixed. J. ACM 31, 114–127 (1984)MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Stojmenovic, I., Tosic, R.: Digitization schemes and the recognition of digital straight lines, hyperplanes and flats in arbitrary dimensions. Vision Geometry, Contemporary Mathematics Series 119, 197–212 (1991)MathSciNetGoogle Scholar
  5. 5.
    Debled-Rennesson, I.: Etude et reconnaissance des droites et plans discrets. PhD Thesis, Université Louis Pasteur, Strasbourg (1995)Google Scholar
  6. 6.
    Debled-Rennesson, I., Reveillès, J.-P.: A new approach to digital planes. In: Proc. Vision Geometry III, SPIE, vol. 2356, pp. 12-21 (1994)Google Scholar
  7. 7.
    Preparata, F.P., Shamos, M.I.: Computational Geometry: An Introduction. Springer, New York (1985)Google Scholar
  8. 8.
    Vittone, J., Chassery, J.-M.: Recognition of digital naive planes and polyhedrization. In: Nyström, I., Sanniti di Baja, G., Borgefors, G. (eds.) DGCI 2000. LNCS, vol. 1953, pp. 296–307. Springer, Heidelberg (2000)CrossRefGoogle Scholar
  9. 9.
    Francon, J., Schramm, J.M., Tajine, M.: Recognizing arithmetic straight lines and planes. In: Miguet, S., Ubéda, S., Montanvert, A. (eds.) DGCI 1996. LNCS, vol. 1176, pp. 141–150. Springer, Heidelberg (1996)Google Scholar
  10. 10.
    Gerard, Y., Debled-Rennesson, I., Zimmermann, P.: An elementary digital plane recognition algorithm 151(1-3), 169–183 (2005)Google Scholar
  11. 11.
    Reveillès, J.-P.: Combinatorial pieces in digital lines and planes. In: Proc. Vision Geometry IV, SPIE, vol. 2573, pp. 23–34 (1995)Google Scholar
  12. 12.
    Veelaert, P.: Digital planarity of rectangular surface segments. IEEE Trans. Pattern Analysis Machine Intelligence 16, 647–652 (1994)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Lilian Buzer
    • 1
  1. 1.Laboratory CNRS-UMLV-ESIEEUMR 8049, ESIEENoisy le GrandFrance

Personalised recommendations