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Parametrization of closed surfaces for 3-D shape description

C. Brechbühler, G. Gerig and O. Kübler
Computer Vision and Image Understanding
Vol. 61, No. 2, pp. 154--170, March 1995


This paper presents procedures for the explicit parametric representation and global description of surfaces of simply connected 3-D objects. The novel techniques overcome severe limitations of earlier methods (restriction to star-shaped objects \cite{BaBr81}, constraints on positioning and shape of cross-sections \cite{Soli90,StDu92}, and nonhomogeneous distribution of parameter space). We parametrize the surface by defining a continuous, one-to-one mapping from the surface of the original object to the surface of a unit sphere. The parametrization is formulated as a constrained optimization problem. Practicable starting values are obtained by an initial mapping based on a heat conduction model. The parametrization enables us to expand the object surface into a series of spherical harmonic functions, extending the concept of elliptical Fourier descriptors for 2-D closed curves [Persoon&Fu77, Kuhl&Giardina82]. Invariant, object-centered descriptors are obtained by rotating the parameter net and the object into standard positions. The new methods are illustrated with simple 3-D test objects. Potential applications are recognition, classification and comparison of convoluted surfaces or parts of surfaces of 3-D shapes.

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  author = {C. Brechb\"uhler and G. Gerig and O. K\"ubler},
  title = {Parametrization of closed surfaces for 3-D shape description},
  journal = {Computer Vision and Image Understanding},
  year = {1995},
  month = {March},
  pages = {154--170},
  volume = {61},
  number = {2},
  keywords = {classification, fitting, matching, morphometry, non-linear, optimization, model-based, recognition, registration, deformable, CT imaging, MR imaging, parametric, shape, surface, volume}