Parameterization of closed surfaces for parametric surface description

A procedure for the parameterization of surface meshes of objects with spherical topology is presented. The generation of such a parameterisation has been formulated and solved as a large constrained optimization problem by C. Brechbuhler (1995), but the convergence of this algorithm becomes unstable for object meshes consisting of several thousand vertices. We propose a new more stable algorithm to overcome this problem using multi-resolution meshes. A triangular mesh is mapped to a sphere by harmonic mapping. Next, a mesh hierarchy is constructed. The coarsest level is then optimized using a modification of the original procedure to map object surfaces to the unit sphere. The result is used as a starting point for the mapping of the next finer mesh, a process which is repeated until the final result is obtained. The new approach is compared to the original one and some parameterized object surfaces are presented.