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Euclidean Nets: An Automatic and Reversible Geometric Smoothing of Discrete 3D Object Boundaries

  • Achille J. -P. Braquelaire
  • Arnault Pousset
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1953)

Abstract

In this work we describe a geometric method to smooth the boundary of a discrete 3D object. The method is reversible in the sense that the discrete boundary can be retrieved by digitizing the smoothed one. To this end, we propose a representation of the boundary of a discrete volume that we call Euclidean net and which is a generalization to the three-dimensional space of Euclidean Path introduced by Braquelaire and Vialard [4]. Euclidean nets can be associated either to voxel based boundaries or to inter-voxel based boundaries. In this paper we focus on the first approach.

Keywords

Digital Surfaces Discrete boundary representation Smoothing 

References

  1. 1.
    J. P. Braquelaire, L. Brun, and A. Vialard. Inter-pixel euclidean paths for image analysis. In Discrete Geometry for Computer Imagery, Lecture Notes in Computer Science 1176, pages 193–204. Springer Verlag, November 1996. 199Google Scholar
  2. 2.
    J. P. Braquelaire, P. Desbarats, J. P. Domenger, and C. A. Wüthrich. A topological structuring for aggregates of 3D discrete objects. In Proc. of the 2nd IAPR-TC-15 Workshop on Graph-based Representation, pages 193–202. Österreichische Computer Gesellschaft, 1999. ISBN 3-85804-126-2. 207Google Scholar
  3. 3.
    J. P. Braquelaire and A. Vialard. A new antialiasing approach for image compositing. The Visual Computer, 13(5):218–227, 1997. 199CrossRefGoogle Scholar
  4. 4.
    J. P. Braquelaire and A. Vialard. Euclidean paths: a new representation of boundary of discrete regions. Graphical Models and Images Processing, 61:16–43, 1999. 198, 199, 201zbMATHCrossRefGoogle Scholar
  5. 5.
    J. M. Chassery and J. Vittone. Coexistence of tricubes in digital naive plane. In Lecture Notes in Computer Science, volume 1347, pages 99–110, December 1997. 203Google Scholar
  6. 6.
    Isabelle Debled. Etude et reconnaissance des droites et plans discrets. Phd thesis, Université Louis Pasteur, Strasbourg, France, 1995. 202, 203zbMATHGoogle Scholar
  7. 7.
    T. J. Fan, G. Medioni, and R. Nevata. Recognising 3D objects using surface descriptions. IEEE Transactions on Pattern Analisys and Machine Intelligence, 1111:1140–1157, 1989. 199CrossRefGoogle Scholar
  8. 8.
    J. Franccon. Discrete combinatorial surfaces. Grapical Models and Image Processing, 57(1):20–26, 1995. 199CrossRefGoogle Scholar
  9. 9.
    Henri Gouraud. Continuous shading of curved surfaces. IEEE Transactions on Computers, C-20(6):623–629, June 1971. 199CrossRefGoogle Scholar
  10. 10.
    G. T. Herman and H. K. Liu. Three-dimensionnal display of uman organs from computed tomograms. Computer Graphics and Image Processing 9, 1:1–21, 1979. 198CrossRefGoogle Scholar
  11. 11.
    K. H. Hohne, M. Bomans, A. Pommert, M. Riemer, C. Schiers, U. Tiede, and G. Wiebecke. 3D visualization of tomographic volume data using the generalized voxel model. The Visual Computer, 6(1):28–36, 1990. 198CrossRefGoogle Scholar
  12. 12.
    J.-O. Lachaud and A. Montanvert. Deformable meshes with automated topology changes for coarse-to-fine 3D surface extraction. Medical Image Analysis, 3(2):187–207, 1999. 199CrossRefGoogle Scholar
  13. 13.
    W. E. Lorenson and H. E. Cline. Marching cubes: A high resolution 3D surface construction algorithm. Computer Graphics (SIGGRAPH’87), 21(4):111–118, 1987. 198CrossRefGoogle Scholar
  14. 14.
    R. Malgouyres. A definition of surfaces of Z3: A new discrete jordan theorem. Theoretical Computer Sciences, 186:6, 1997. 199Google Scholar
  15. 15.
    FD. G. Morgenthaler and A. Rosenfeld. Surfaces in three-dimensional digital images. Information and Control, 51:227–247, 1981. 199zbMATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    J. P. Reveilles. G’eométrie discréte, Calcul en nombres entiers et algorithmique. PhD thesis, Université de Strasbourg, 1991. 201, 202Google Scholar
  17. 17.
    A. Rosenfeld, T.Yung Kong, and A. Y. Wu. Digital surfaces. Grapical Models and Image Processing, 53(4):305–312, 1991. 199zbMATHCrossRefGoogle Scholar
  18. 18.
    J. M. Schramm. Coplanar tricubes. In Springer-Verlag, editor, Lecture Notes in Computer Science, volume 1347, pages 87–98, 1997. 203CrossRefGoogle Scholar
  19. 19.
    P. Tellier and I. Debled-Rennesson. 3d discrete normal vectors. In Springer-Verlag, editor, Lecture Notes in Computer Science, volume 1568, pages 447–458, 1999. 205Google Scholar
  20. 20.
    D. Terzopoulos, A. Witkin, and M. Kass. Symmetry-seeking models and 3D object reconstruction. International Journal of Computer Vision, 1(3):211–221, 1987. 199CrossRefGoogle Scholar
  21. 21.
    D. Terzopoulos, A. Witkin, and M. Kass. Constraints on deformable models: Recovering 3D shape and nonrigid motion. Artificial Intelligence, 36(1):91–123, 1988. 199zbMATHCrossRefGoogle Scholar
  22. 22.
    G. Thürmer. Surfaces in Three-Dimensional Discrete Space. PhD thesis, Fakultät Medien der Bauhaus-Universität Weimar, 1998. Shaker Verlag. 199, 201Google Scholar
  23. 23.
    G. Thürmer and C. A. Wüthrich. Normal computation for discrete surfaces in 3D space. Computer Graphics Forum, 16(3):15–26, 1997. Proceedings of Eurographics’ 97. 205CrossRefGoogle Scholar
  24. 24.
    G. Thürmer and C. A. Wüthrich. Polygon mesh generation for discrete surfaces in 3d space. In W. Lefer and M. Grave, editors, Proc. of the Eighth Eurographics Workshop on Visualization in Scientific Computing, pages 117–126, Laboratoire d’Informatique du Littoral, Boulogne sur Mer (France), 1997. 199Google Scholar
  25. 25.
    A. Vialard. Geometrical parameters extraction from discrete paths. In Discrete Geometry for Computer Imagery, Lecture Notes in Computer Science 1176, pages 24–35. Springer Verlag, November 1996. 199, 207Google Scholar
  26. 26.
    A. Vialard. Euclidean paths for representing and transforming scanned characters. In Proceedings of GREC’97. Second IAPR Workshop on Graphics Recognition, 1997. To be published in Lecture Notes in Computer Science, Springer Verlag. 199Google Scholar
  27. 27.
    Anne Vialard. Chemins euclidiens: Un modéle de représentation des contours discrets. Phd thesis, Université Bordeaux 1, 1996. 199Google Scholar
  28. 28.
    J. Vittone. Caractérisation et reconnaissance de droites et de plans en géométrie discrète. Phd thesis, Université de Grenoble, 1999. 203Google Scholar
  29. 29.
    J. Vittone and J. M. Chassery. (n,m)-cubes and farey nets for digital naive planes understanding. In Lecture Notes in Computer Science, volume 1568, pages 76–87, 1999. 203, 204Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2000

Authors and Affiliations

  • Achille J. -P. Braquelaire
    • 1
  • Arnault Pousset
  1. 1.LaBRI - Laboratoire Bordelais de Recherche en Informatique - UMR 5800Université Bordeaux 1TalenceFrance

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