Recovering Surfaces from the Restoring Force

  • George Kamberov
  • Gerda Kamberova
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2351)


We present a new theoretical method and experimental results for direct recovery of the curvatures, the principal curvature directions, and the surface itself by explicit integration of the Gauss map. The method does not rely on polygonal approximations, smoothing of the data, or model fitting. It is based on the observation that one can recover the surface restoring force from the Gauss map, and (i) applies to orientable surfaces of arbitrary topology (not necessarily closed); (ii) uses only first order linear differential equations; (iii) avoids the use of unstable computations; (iv) provides tools for filtering noise from the sampled data. The method can be used for stable extraction of surfaces and surface shape invariants, in particular, in applications requiring accurate quantitative measurements.


  1. [1]
    Bajcsy, R., Euciso, R., Kamberova, G., Nocera, L., Šara, R.: 3D Reconstruction of Environments for Virtual Collaboration, Proc. 4th IEEE Workshop on Applications of Computer Vision, Princeton, NJ,(1998).Google Scholar
  2. [2]
    Boissonnat, J.D., Cazals, F.: Smooth surface reconstruction via natural neighbour interpolation of distance functions. In ACM Proc XVIth Annl Symp. in Comp. Geom. Hong Kong (2000).Google Scholar
  3. [3]
    Besl, P., Jain, R.: Invariant surface characteristics and 3d object recognition in range images. CVGIP, 33:33–80, (1986).Google Scholar
  4. [4]
    Quicken, M., Brechbüller, C., et al: Parameterization of Closed Surfaces for Parametric Surface Description. Proceedings IEEE CVPIR 2000, volume 1, 354–360.Google Scholar
  5. [5]
    DoCarmo, M.: Differential Geometry of Curves and Surfaces. Prentice-Hall, (1976).Google Scholar
  6. [6]
    Devernay, F.: “Computing Differential Properties of 3-D Shapes from Stereoscopic Images without 3-D Models”, INRIA, RR-2304, Sophia Antipolis, (1994).Google Scholar
  7. [7]
    Feldmar, J., Ayache, N.: Registration of Smooth Surfaces Using Differential Properties. Number 801. Springer-Verlag, (1994).Google Scholar
  8. [8]
    Ikeuchi, K., Herbert, M.: Spherical Representations: from EGI to SAI CMU Technical report: CMU-CS-95-197 (1995).Google Scholar
  9. [9]
    Kamberov, G., Norman, P., Pinkall, U., Pedit, F.: Quaternionions, Spinors, and Surfaces to appear in Contemporary Mathematics, AMSGoogle Scholar
  10. [10]
    Kamberov, G., Kamberova, G.: Shape Invariants and Principal Directions from 3D Points and Normals, Proceedings of 10th Intl. Conf. in Central Europe on Computer Graphics and Visualization, (2002), Plzen. Journal of WSCG, Volume 10, (2002), Pages 537–544.Google Scholar
  11. [11]
    Kobayashi, S., Nomidzu, K.: Foundations of Differential Geometry. Volume 2. Chapter IX. Interscience Publ. New York. (1969).zbMATHGoogle Scholar
  12. [12]
    Koenderink, J.: Solid Shape MIT press, (1990).Google Scholar
  13. [13]
    Metaxas, D., Terzpoulos, D.: Dynamic 3D models with local and global deformations: deformable superquadrics, IEEE PAMI, 13(7):703–714, (1991).Google Scholar
  14. [14]
    Šára, R., Bajcsy, R.: “Fish-Scales: Representing Fuzzy Manifolds,” Proc. Int. Conference on Computer Vision, Bombay, India, Narosa Publishing House, (1998).Google Scholar
  15. [15]
    Shum, H., Herbert, M., Ikeuci, K.: On 3d shape synthesis. In Proc. Image Under. Workshop, volume 2, pages 1103–1112, (1996).Google Scholar
  16. [16]
    Sommerfeld, A.: Mechanics of Defeormable Bodies. Lectures in Theoretical Physics volume 2, pp 122–124, Academic Press, New York, (1950).Google Scholar
  17. [17]
    Taubin, G.: Estimating the tensor of curvature of a surface from a polyhedral approximation. In Proc. ICCV. IEEE Comp. Press, (1995).Google Scholar
  18. [18]
    Worthington, P., Hancock, E.: Histogram-based Object Recognition Using Shape from Shading. Proc. IEEE Conf. CVPR 2000, volume 1, 643–648 (2000).Google Scholar
  19. [19]
    Yuen, P., et al: Curvature and Torsion Feature Extraction from 3-D Meshes and Multiple Scales. IEEE Proc. Vis. Image Signal Process, volume 147, No 5, 454–462 (2000).CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2002

Authors and Affiliations

  • George Kamberov
    • 1
  • Gerda Kamberova
    • 2
  1. 1.Stevens Institute of TechnologyHobokenUSA
  2. 2.Hofstra UniversityHempsteadUSA

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