Fast Spherical Mapping for Genus-0 Meshes

Parameterizing a genus-0 mesh onto a unit sphere means assigning a 3D position on the unit sphere to each vertex of the mesh, such that the spherical mapping induced by the mesh connectivity is not too distorted and does not have overlapping areas. The non-overlapping requirement is technically the most difficult component also the most critical component of many spherical parametrization methods. In this paper we propose a fast spherical mapping approach that can map any closed genus-0 mesh onto a unit sphere without overlapping any part of the given mesh. The mapping process does not require setting up any linear systems, nor any expensive matrix computation, but is done simply by iteratively moving vertices of the given mesh locally until a desired spherical mapping is reached. Therefore the new spherical mapping approach is very fast and, consequently, can be used for meshes with large number of vertices. Moreover, the iterative process is guaranteed to converge. Another interesting phenomenon about this new approach is, it can generate meaningful results without considering the angle-preserving or edge-length-preserving constraint in the mapping process. Our apporach can be used for texture mapping, remeshing, 3D morphing and, more importantly, can be used as input for other more rigourous and expensive spherical parametrization methods to achieve more accurate parametrization results. Several test results are included to demonstrate the new apporach’s capability in performing spherical mapping without any overlapping.