Implicit Digital Surfaces in Arbitrary Dimensions

  • Jean-Luc Toutant
  • Eric Andres
  • Gaelle Largeteau-Skapin
  • Rita Zrour
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8668)


In this paper we introduce a notion of digital implicit surface in arbitrary dimensions. The digital implicit surface is the result of a morphology inspired digitization of an implicit surface {x ∈ ℝn : f(x) = 0} which is the boundary of a given closed subset of ℝ n , {x ∈ ℝn : f(x) ≤ 0}. Under some constraints, the digital implicit surface has some interesting properties, such as k-tunnel freeness. Furthermore, for a large class of the digital implicit surfaces, there exists a very simple analytical characterization.


Implicit Curve Implicit Surface Digital Object Flake Digitization 


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  1. 1.
    Bloomenthal, J., Wyvill, B. (eds.): An Introduction to Implicit Surfaces. Morgan Kaufmann Publishers Inc., San Francisco (1997)Google Scholar
  2. 2.
    Velho, L., Gomes, J., de Figueiredo, L.H. (eds.): Implicit Objects in Computer Graphics. Springer (2002)Google Scholar
  3. 3.
    Stolte, N.: Arbitrary 3D resolution discrete ray tracing of implicit surfaces. In: Andrès, É., Damiand, G., Lienhardt, P. (eds.) DGCI 2005. LNCS, vol. 3429, pp. 414–426. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  4. 4.
    Emeliyanenko, P., Berberich, E., Sagraloff, M.: Visualizing arcs of implicit algebraic curves, exactly and fast. In: Bebis, G., et al. (eds.) ISVC 2009, Part I. LNCS, vol. 5875, pp. 608–619. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  5. 5.
    Stolte, N., Caubet, R.: Comparison between Different Rasterization Methods for Implicit Surfaces. In: Visualization and Modeling, ch. 10. Academic Press (1997)Google Scholar
  6. 6.
    Stolte, N., Kaufman, A.: Novel techniques for robust voxelization and visualization of implicit surfaces. Graphical Models 63(6), 387–412 (2001)CrossRefzbMATHGoogle Scholar
  7. 7.
    Sigg, C.: Representation and Rendering of Implicit Surface. Phd thesis, diss. eth no. 16664, ETH Zurich, Switzerland (2006)Google Scholar
  8. 8.
    Taubin, G.: An accurate algorithm for rasterizing algebraic curves. In: Proceedings of the 2nd ACM Solid Modeling and Applications, pp. 221–230 (1993)Google Scholar
  9. 9.
    Taubin, G.: Rasterizing algebraic curves and surfaces. IEEE Computer Graphics and Applications 14(2), 14–23 (1994)CrossRefGoogle Scholar
  10. 10.
    Toutant, J.-L., Andres, E., Roussillon, T.: Digital circles, spheres and hyperspheres: From morphological models to analytical characterizations and topological properties. Discrete Applied Mathematics 161(16-17), 2662–2677 (2013)CrossRefzbMATHMathSciNetGoogle Scholar
  11. 11.
    Andres, E.: The supercover of an m-flat is a discrete analytical object. Theoretical Computer Science 406(1-2), 8–14 (2008)CrossRefzbMATHMathSciNetGoogle Scholar
  12. 12.
    Cohen-Or, D., Kaufman, A.E.: Fundamentals of surface voxelization. CVGIP: Graphical Model and Image Processing 57(6), 453–461 (1995)Google Scholar
  13. 13.
    Stelldinger, P., Köthe, U.: Towards a general sampling theory for shape preservation. Image and Vision Computing 23(2), 237–248 (2005)CrossRefGoogle Scholar
  14. 14.
    Brimkov, V.E., Andres, E., Barneva, R.P.: Object discretizations in higher dimensions. Pattern Recognition Letters 23(6), 623–636 (2002)CrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  • Jean-Luc Toutant
    • 1
  • Eric Andres
    • 2
  • Gaelle Largeteau-Skapin
    • 2
  • Rita Zrour
    • 2
  1. 1.Clermont Université, Université d’Auvergne, ISIT, UMR CNRS 6284Clermont-FerrandFrance
  2. 2.Université de Poitiers, Laboratoire XLIM, SIC, UMR CNRS 7252Futuroscope ChasseneuilFrance

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