Towards Representation of 3D Shape: Global Surface Parametrization

Abstract

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.