Parameterizations of digital surfaces homeomorphic to a sphere using discrete harmonic functions

Abstract

We discuss how to map simply connected digital surfaces to the unit sphere using discrete harmonic functions. This technique is well known for constructing parametrizations of surfaces onto simpler domains. A common problem is the creation of dense clusters of vertices, which leads to numerical instability of methods operating on this mapping. By an explicit calculation, we quantify the cluster density for a simple model example. This example shows that distances between mapped vertices can decrease exponentially in regions on the sphere. By numerical examples, we show that severe clustering often occurs for natural objects. A computationally inexpensive algorithm, based on a bijective transformation of the unit sphere, is suggested for post-processing of clusters. Experiments indicate that this algorithm often improves the mapping such that convergence of a non-linear optimization program is achieved. The program aims at optimizing the parameterization, making it useful for global shape analysis. Examples of approximations in terms of spherical harmonics functions are presented.