Uniform Parameterization of Triangulated Spherical Topology Surfaces for Invariant 3-D Shape Description

A. Ben Abdallah and F. Ghorbel (Tunisia)


Spherical harmonics model, 3D reconstruction, 3D shape descriptors, surface parameterization, invariant shape descriptor, harmonic analysis.


This paper presents a new algorithm for uniform parameterization and global description of 3D triangular mesh surfaces of objects with spherical topology. Our approach is decomposed into two steps. The first step consists in initially parameterize the surface by defining a continuous one-to-one mapping from the surface of the object to the surface of a unit sphere based on a heat conduction model [4]. The optimization of initial parameterization is formulated as a constrained optimization problem. The second step consists in expanding the surface into a series of spherical harmonic functions. The coefficients of the series are computed by solving a least-squares problem. The obtained coefficients depend on the relative position of the parameter net of the object surface and on the orientation of the object in space. To overcome this limitation, we propose a solution to transform rotation dependent shape descriptor into rotation independent one based on abstract harmonic analysis and using results from representation theory. The Fourier Transform is generalized to any homogenous space with a transitively acting group. Such a case is the unit sphere 2 S with rotation group )3(SO as the acting group. The shift theorem permits us to extract a rotation invariant shape descriptors.

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