Multiresolution Hierarchies on Unstructured Triangle Meshes

The use of polygonal meshes for the representation of highly complex geometric objects has become the de facto standard in most computer graphics applications. Especially triangle meshes are preferred due to their algorithmic simplicity, numerical robustness, and efficient display. The possibility to decompose a given triangle mesh into a hierarchy of differently detailed approximations enables sophisticated modeling operations like the modification of the global shape under preservation of the detail features. So far, multiresolution hierarchies have been proposed mainly for meshes with subdivision connectivity. This type of connectivity results from iteratively applying a uniform split operator to an initially given coarse base mesh. In this paper we demonstrate how a similar hierarchical structure can be derived for arbitrary meshes with no restrictions on the connectivity. Since smooth (subdivision) basis functions are no longer available in this generalized context, we use constrained energy minimization to associate smooth geometry with coarse levels of detail. As the energy minimization requires one to solve a global sparse system, we investigate the effect of various parameters and boundary conditions in order to optimize the performance of iterative solving algorithms. Another crucial ingredient for an effective multiresolution decomposition of unstructured meshes is the flexible representation of detail information. We discuss several approaches.