Visual Form pp 79-88 | Cite as

Towards Representation of 3D Shape: Global Surface Parametrization

  • Ch. Brechbühler
  • G. Gerig
  • O. Kübler


A procedure for the parametrization of the surface of a simply connected object is presented. Starting from a relational data structure describing surface nodes and links to edges and vertices, a distance transform is applied to determine two distant poles. The physical model of a heat conducting surface is then used to obtain latitude and longitude parameters. The net created assigns a unique coordinate pair to each surface node, but its structure depends on the selection of the poles and comprises a systematic nonuniformity of node distributions over the sphere. To correct distortions and to achieve independence of starting conditions, an isotropic non-linear relaxation of the node locations on the sphere is developed. This dynamic modelling procedure is used to obtain the final parametrization.

The advantage of the proposed technique is a unique mapping of arbitrary simply connected, but not necessarily convex or star-shaped, objects to spherical coordinates. The example demonstrates the uniform distribution of nodes projected onto the sphere and the object symmetries that are reproduced. The mapping of a surface to a sphere represents a prerequisite processing stage to the representation of 3D shape in terms of an elliptic harmonic decomposition and is illustrated by a simple example.


North Pole South Pole Original Object Edge Node Surface Node 
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]
    H. Asada and M. Brady, The curvature primal sketch, PAMI-8, No.1, 2-14 (January 1986).Google Scholar
  2. [2]
    F.P. Kuhl and Ch.R. Giardina, Elliptic Fourier Features of a Closed Contour, CGIP 18, 236-258 (1982).Google Scholar
  3. [3]
    J.K. Udupa, Boundary and Object Labelling in Three-Dimensional Images, CVGIP 51, 355-369 (1990).Google Scholar
  4. [4]
    P.T. Sander and St. W. Zucker, Inferring Surface Trace and Differential Structure from 3-D Images, IEEE PAMI-12, No. 9, 833-854 (September 1990).Google Scholar
  5. [5]
    H. Yamada, Complete Euclidean distance transformations by parallel operation, Proc. 7.ICPR, Montreal Canada, pp. 69-71 (1984).Google Scholar
  6. [6]
    W. H. Press, B. P. Flannery, S. A. Teukolsky and W. Vetterling, Numerical Recipes in C — The Art of Scientific Computing, Cambridge University Press, Cambridge, p.636, (1988).zbMATHGoogle Scholar
  7. [7]
    L. Nirenbarg, A strong maximum principle for parabolic equations, Commun. Pure Appl. Math., vol. VI, pp. 167–177 (1953).CrossRefGoogle Scholar
  8. [8]
    W. Greiner and H. Diehl, Theoretische Physik — Ein Lehr-und Übungsbuch für Anfangssemester, Band 3: Elektrodynamik; Verlag Harri Deutsch, Zürich und Frankfurt am Main; pp. 61-65.Google Scholar

Copyright information

© Springer Science+Business Media New York 1992

Authors and Affiliations

  • Ch. Brechbühler
    • 1
  • G. Gerig
    • 1
  • O. Kübler
    • 1
  1. 1.Institute for Communication Technology, Image Science DivisionETH-ZentrumZurichSwitzerland

Personalised recommendations