Computational Aspects of Digital Plane and Hyperplane Recognition

  • David Coeurjolly
  • Valentin Brimkov
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4040)


In these note we review some basic approaches and algorithms for discrete plane/hyperplane recognition. We present, analyze, and compare related theoretical and experimental results and discuss on the possibilities for creating algorithms with higher efficiency.


Grid Point Convex Hull Recognition Algorithm Euclidean Plane Computational Aspect 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
  2. 2.
    Acketa, D.M., Žunić, J.D.: On the maximal number of edges of convex digital polygons included into a m×m-grid. Journal of Combinatorial Theory Serie A(69), 358–368 (1995)Google Scholar
  3. 3.
    Andres, E., Acharya, R., Sibata, C.: Discrete analytical hyperplanes. Graphical Models and Image Processing 59(5), 302–309 (1997)CrossRefGoogle Scholar
  4. 4.
    Avis, D.: lrs: implementation as a callable library of the reverse search algorithm for vertex enumeration/convex hull problems,
  5. 5.
    Balog, A., Bárány, I.: On the convex hull of the integer points in a disc. In: ACM -SIGACT ACM-SIGGRAPH (ed.) Proceedings of the 7th Annual Symposium on Computational Geometry (SCG 1991), North Conway, NH, USA, June 1991, pp. 162–165. ACM Press, New York (1991)CrossRefGoogle Scholar
  6. 6.
    Barany, I.: Random points, convex bodies, lattices. In: ICM: Proceedings of the International Congress of Mathematicians (2002)Google Scholar
  7. 7.
    Barany, I., Howe, Lovasz: On integer points in polyhedra: A lower bound. Combinatorica 12 (1992)Google Scholar
  8. 8.
    Bárány, I., Larman, D.G.: The convex hull of the integer points in a large ball. Math. Annalen 312, 167–181 (1998)MATHCrossRefGoogle Scholar
  9. 9.
    Barber, C.B., Dobkin, D.P., Huhdanpaa, H.: The quickhull algorithm for convex hulls. ACM Transactions on Mathematical Software 22(4), 469–483 (1996)MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Barneva, R.P., Brimkov, V.E., Nehlig, P.: Thin discrete triangular meshes. Theoretical Computer Science 246(1-2), 73–105 (2000)MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Bremner, D.: Incremental convex hull algorithms are not output sensitive. Discrete & Computational Geometry 21(1), 57–68 (1999)MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Brimkov, V., Coeurjolly, D., Klette, R.: Digital planarity - a review. Technical report, Laboratoire LIRIS, Université Claude Bernard Lyon 1 (2004),
  13. 13.
    Brimkov, V.E., Dantchev, S.S.: Complexity analysis for digital hyperplane recognition in arbitrary fixed dimension. In: Andrès, É., Damiand, G., Lienhardt, P. (eds.) DGCI 2005. LNCS, vol. 3429, pp. 287–298. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  14. 14.
    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
  15. 15.
    Coeurjolly, D.: Algorithmique et géométrie discrète pour la caractérisation des courbes et des surfaces. PhD thesis, Université Lumière Lyon 2, Bron, Laboratoire ERIC (December 2002)Google Scholar
  16. 16.
    Coeurjolly, D., Guillaume, A., Sivignon, I.: Reversible discrete volume polyhedrization using marching cubes simplification. In: SPIE Vision Geometry XII, San Jose, USA, vol. 5300, pp. 1–11 (2004)Google Scholar
  17. 17.
    Coeurjolly, D., Sivignon, I., Dupont, F., Feschet, F., Chassery, J.-M.: On digital plane preimage structure. Discrete Applied Mathematics 151(1–3), 78–92 (2005)MATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Debled-Rennesson, I.: Etude et reconnaissance des droites et plans discrets. PhD thesis, Université Louis Pasteur (1995)Google Scholar
  19. 19.
    Françon, 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
  20. 20.
    Françon, J., Papier, L.: Polyhedrization of the boundary of a voxel object. In: Bertrand, G., Couprie, M., Perroton, L. (eds.) DGCI 1999. LNCS, vol. 1568, pp. 425–434. Springer, Heidelberg (1999)CrossRefGoogle Scholar
  21. 21.
    Garey, M.R., Johnson, D.S.: Computers and intractability; a guide to the theory of NP-completeness. W.H. Freeman, New York (1979)MATHGoogle Scholar
  22. 22.
    Gerard, Y., Debled-Rennesson, I., Zimmermann, P.: An elementary digital plane recognition algorithm. DAMATH: Discrete Applied Mathematics and Combinatorial Operations Research and Computer Science 151 (2005)Google Scholar
  23. 23.
    Kim, C.E.: Three-dimensional digital planes. IEEE Trans. on Pattern Analysis and Machine Intelligence 6, 639–645 (1984)MATHCrossRefGoogle Scholar
  24. 24.
    Kim, C.E., Stojmenovic, I.: On the recognition of digital planes in three-dimensional space. Pattern Recognition Letters 12(11), 665–669 (1991)CrossRefGoogle Scholar
  25. 25.
    Klette, R., Stojmenovic, I., Zunic, J.D.: A parametrization of digital planes by least-squares fits and generalizations. CVGIP: Graphical Model and Image Processing 58(3), 295–300 (1996)CrossRefGoogle Scholar
  26. 26.
    Klette, R., Sun, H.J.: Digital planar segment based polyhedrization for surface area estimation. In: Arcelli, C., Cordella, L.P., Sanniti di Baja, G. (eds.) IWVF 2001. LNCS, vol. 2059, pp. 356–366. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  27. 27.
    Megiddo, N.: Linear programming in linear time when the dimension is fixed. JACM: Journal of the ACM 31 (1984)Google Scholar
  28. 28.
    Mesmoudi, M.M.: A simplified recognition algorithm of digital planes pieces. In: Braquelaire, A., Lachaud, J.-O., Vialard, A. (eds.) DGCI 2002. LNCS, vol. 2301, pp. 404–416. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  29. 29.
    Preparata, F.P., Shamos, M.I.: Computational Geometry: An Introduction. Springer, Heidelberg (1985)Google Scholar
  30. 30.
    Rosenfeld, A.: Digital straight lines segments. IEEE Transactions on Computers, 1264–1369 (1974)Google Scholar
  31. 31.
    Rosenfeld, A., Klette, R.: Digital straightness. In: Int. Workshop on Combinatorial Image Analysis. Electronic Notes in Theoretical Computer Science, vol. 46. Elsevier Science Publishers, Amsterdam (2001)Google Scholar
  32. 32.
    Schrijver, A.: Theory of Linear and Integer Programming. Wiley and Sons, Chichester (1986)MATHGoogle Scholar
  33. 33.
    Sivignon, I., Coeurjolly, D.: From digital plane segmentation to polyhedral representation. In: Asano, T., Klette, R., Ronse, C. (eds.) Geometry, Morphology, and Computational Imaging. LNCS, vol. 2616, pp. 356–367. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  34. 34.
    Sivignon, I., Dupont, F., Chassery, J.M.: Decomposition of a three-dimensional discrete object surface into discrete plane pieces. Algorithmica 38(1), 25–43 (2003)CrossRefMathSciNetGoogle Scholar
  35. 35.
    Stojmenović, I., Tosić, R.: Digitization schemes and the recognition of digital straight lines, hyperplanes and flats in arbitrary dimensions. In: Vision Geometry, contemporary Mathematics Series, vol. 119, pp. 197–212. American Mathematical Society, Providence (1991)Google Scholar
  36. 36.
    Veelaert, P.: On the flatness of digital hyperplanes. Journal of Mathematical Imaging and Vision 3, 205–221 (1993)MATHCrossRefMathSciNetGoogle Scholar
  37. 37.
    Veelaert, P.: Digital planarity of rectangular surface segments. IEEE Pattern Analysis and Machine Intelligence 16(6), 647–652 (1994)CrossRefGoogle Scholar
  38. 38.
    Vittone, J., Chassery, J.-M.: Recognition of digital naive planes and polyhedization. In: Nyström, I., Sanniti di Baja, G., Borgefors, G. (eds.) DGCI 2000. LNCS, vol. 1953, pp. 296–307. Springer, Heidelberg (2000)CrossRefGoogle Scholar
  39. 39.
    Zolotykh, N.Y.: On the number of vertices in integer linear programming problems. Technical report, University of Nizhni Novgorod (2000)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • David Coeurjolly
    • 1
  • Valentin Brimkov
    • 2
  1. 1.Laboratoire LIRIS – CNRS UMR 5205Université Claude Bernard Lyon1VilleurbanneFrance
  2. 2.Mathematics Department, Buffalo State CollegeState University of New YorkBuffaloUSA

Personalised recommendations