Shape Reconstruction by Line Voting in Discrete Space

  • Kosuke Sato
  • Atsushi Imiya
  • Tomoya Sakai
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4291)


Shape from silhouettes is a binary geometric tomography since both objects and projections, which are measured as silhouettes, are binary. In this paper, we formulate shape from silhouettes in the three-dimensional discrete space. This treatment of the problem implies an ambiguity theorem for the reconstruction of objects in discrete space. Furthermore, we show that in three-dimensional space, it is possible to reconstruct a class of non-convex objects from a collection of silhouettes though on a plane non-convex object is unreconstractable from any collection of silhouettes.


Convex Body Convex Polygon Discrete Space Perspective Projection Visible Hull 
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.
    Guggenheimer, H.W.: Applicable Geometry. Robert E. Kniegen Pub, Inci, New York (1977)MATHGoogle Scholar
  2. 2.
    Aloimonos, J.: Visual shape computation. Proceedings of IEEE 76, 899–916 (1988)CrossRefGoogle Scholar
  3. 3.
    Dobkin, D.P., Edelsbrunner, H., Yap, C.K.: Probing convex polytopes. In: Proc. 18th ACM Symposium on Theory of Computing, pp. 424–432 (1986)Google Scholar
  4. 4.
    Campi, S.: Reconstructing a convex surface from certain measurements of its projections, bollettio U.M.I.  6, 945–959 (1986)Google Scholar
  5. 5.
    Boltyanski, V., Martin, H., Soltan, P.S.: Excursions into Combinatorial Geometry. Springer, Berlin (1997)MATHGoogle Scholar
  6. 6.
    Kutulakos, K., Seitz, S.M.: A theory of shape by space carving. In: Proceedings of 7th ICCV, vol. 1, pp. 307–314 (1999)Google Scholar
  7. 7.
    Skiena, S.S.: Interactive reconstruction via geometric probing. IEEE Proceedings 80, 1364–1383 (1992)CrossRefGoogle Scholar
  8. 8.
    Skiena, S.S.: Probing convex polygon with half-planes. Journal of Algorithms 12, 359–374 (1991)MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Lindembaum, M., Bruckstein, A.: Reconstructing a convex polygon from binary perspective projections. Pattern Recognition 23, 1343–1350 (1990)CrossRefMathSciNetGoogle Scholar
  10. 10.
    Laurentini, A.: The visual hull concept for silhouette-bases image understanding. IEEE PAMI 16, 150–163 (1994)Google Scholar
  11. 11.
    Laurentini, A.: How for 3D shape can be understood from 2D silhouettes. IEEE PAMI 17, 88–195 (1995)Google Scholar
  12. 12.
    Li, R.S.-Y.: Reconstruction of polygons from projections. Information Processing Letters 28, 235–240 (1988)MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Prince, J.L., Willsky, A.S.: Reconstructing convex sets from support line measurements. IEEE Trans PAMI 12, 377–389 (1990)Google Scholar
  14. 14.
    Rao, A.S., Goldberg, Y.K.: Shape from diameter: Recognizing polygonal parts with parallel-jaw gripper. International Journal of Robotics Research 13, 16–37 (1994)CrossRefGoogle Scholar
  15. 15.
    Kawamoto, K., Imiya, K.: Detection of spatial points and lines by random sampling and voting process. Pattern Recognition Letters 22, 199–207 (2001)MATHCrossRefGoogle Scholar
  16. 16.
    Solmon, D.C.: The X-ray transform. Journal of Math. Anal. and Appl. 56, 61–83 (1976)MATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Hammaker, C., Smith, K.T., Solomon, D.C., Wagner, L.: The divergent beam x-ray transform. Rocky Mountain Journal of Mathematics 10, 253–283 (1980)CrossRefMathSciNetGoogle Scholar
  18. 18.
    Imiya, A., Kawamoto, K.: Shape reconstruction from an image sequences. In: Arcelli, C., Cordella, L.P., Sanniti di Baja, G. (eds.) IWVF 2001. LNCS, vol. 2059, pp. 677–686. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  19. 19.
    Imiya, A., Kawamoto, K.: Mathematical aspects of shape reconstruction from an image sequence. In: Proc. 1st Intl. Symp. 3D data Processing Visualization and Transformations, pp. 632–635 (2002)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Kosuke Sato
    • 1
  • Atsushi Imiya
    • 2
  • Tomoya Sakai
    • 2
  1. 1.School of Science and TechnologyChiba UniversityJapan
  2. 2.Institute of Media and Information TechnologyChiba University, JapanChibaJapan

Personalised recommendations