Experiments on the Parameterization of Point Clouds with Holes

A meshless method for parameterizing point clouds using convex combination mapping and mean value coordinates has been introduced by Floater et al. The advantage of the method is that only a sparse linear system has to be solved and the resulting parameterization is usually injective. However, in the presence of concave boundaries the convex combination mapping is not always well defined and often results in considerable distortion in the parameterization. We propose a method to extend the meshless parameterization approach, in such a way that distortion is avoided in the vicinity of concave boundaries. To ensure that the convex combination mapping is always defined, virtual points are appended to each concave neighborhood. The algorithm operates directly on point clouds and does not require any triangulation or connectivity information.