Maximizing adaptivity in hierarchical topological models

Abstract

We present an approach to hierarchically encode the topology of functions over triangulated surfaces. Its Morse-Smale complex, a well known structure in computational topology, describes the topology of a function. Following concepts of Morse theory, a Morse-Smale complex (and therefore a function's topology) can be simplified by successively canceling pairs of critical points. We demonstrate how cancellations can be effectively encoded to produce a highly adaptive topology-based multi-resolution representation of a given function. Contrary to the approach, we avoid encoding the complete complex in a traditional mesh hierarchy. Instead, the information is split into a new structure we call a cancellation forest and a traditional dependency graph. The combination of this new structure with a traditional mesh hierarchy proofs to be significantly more flexible than the one previously reported. In particular, we can create hierarchies that are guaranteed to be of logarithmic height.