Three-Dimensional Structure Detection from Anisotropic Alpha-Shapes

  • Sébastien Bougleux
  • Mahmoud Melkemi
  • Abderrahim Elmoataz
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3656)


We present an application of a family of affine diagrams to the detection of three-dimensional sampled structures embedded in a perturbated background. This family of diagrams is an extension of the Voronoi diagram, namely the anisotropic diagrams. These diagrams are defined by using a parameterized distance whose unit ball is an ellipsoidal one. The parameters, upon which depends this distance, control the elongation and the orientation of the associated ellipsoidal ball. Based on these diagrams, we define the three-dimensional anisotropic α-shape concept. This concept is an extension of the Euclidean one, it allows us to detect structures, as straight lines and planes, in a given direction. The detection of a more general polyhedral structure is obtained by merging several anisotropic α-shapes, computed for different orientations.


Voronoi Diagram Linear Structure Delaunay Triangulation Elongation Ratio Anisotropic Mesh 
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  1. 1.
    Lee, D.T., Wong, C.K.: Voronoi diagrams in L 1 (L  ∞ ) metrics with 2-dimensional storage applications. SIAM J. Comput. 9, 200–211 (1980)zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Aurenhammer, F.: Power diagrams: properties, algorithms and applications. SIAM J. Comput. 16, 78–96 (1987)zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Aurenhammer, F., Imai, H.: Geometric relations among Voronoi diagrams. Geometriae Dedicata 27, 65–75 (1988)zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Goodman, J.E., O’Rourke, J.: Handbook of Discrete and Computational Geometry, 2nd edn. Chapman and Hall, Boca Raton (2004)zbMATHCrossRefGoogle Scholar
  5. 5.
    Toussaint, G.T.: Pattern Recognition and Geometrical Complexity. In: Proc. Int. Conf. on Pattern Recognition, December 1980, pp. 1324–1347 (1980)Google Scholar
  6. 6.
    Melkemi, M., Djebali, M.: Elliptic diagrams: application to patterns detection from a finite set of points. Pattern Recognition Letters 22(8), 835–844 (2001)zbMATHCrossRefGoogle Scholar
  7. 7.
    Teichmann, M., Capps, M.: Surface reconstruction with anisotropic density-scaled alpha shapes. In: VIS 1998: Proc. of the conference on Visualization, pp. 67–72. IEEE Computer Society Press, Los Alamitos (1998)Google Scholar
  8. 8.
    Labelle, F., Shewchuk, J.R.: Anisotropic voronoi diagrams and guaranteed-quality anisotropic mesh generation. In: SCG 2003: Proc. of the 9th Annual Symposium on Computational Geometry, pp. 191–200. ACM Press, San Diego (2003)Google Scholar
  9. 9.
    Edelsbrunner, H., Mücke, E.P.: Three-Dimensional Alpha Shapes. ACM Transactions on Graphics 13(1), 43–72 (1994)zbMATHCrossRefGoogle Scholar
  10. 10.
    Duda, R.O., Hart, P.E., Stork, D.G.: Pattern Classification, 2nd edn. Wiley Interscience, Hoboken (2000)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  • Sébastien Bougleux
    • 1
  • Mahmoud Melkemi
    • 2
  • Abderrahim Elmoataz
    • 3
  1. 1.GREYC CNRS UMR 6072, ENSICAENCaen CedexFrance
  2. 2.LMIA, équipe MAGEMulhouse CedexFrance
  3. 3.LUSAC, Site UniversitaireCherbourg-OctevilleFrance

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