Advertisement

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)

Abstract

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.

Keywords

Implicit Curve Implicit Surface Digital Object Flake Digitization 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  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

Personalised recommendations