Maximizing adaptivity in hierarchical topological models

We present an approach to hierarchically encode the topology of functions over triangulated surfaces. We describe the topology of a function by its Morse-Smale complex, a well known structure in computational topology. 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 pro-duce a highly adaptive topology-based multi-resolution representa-tion of a given function. Contrary to the approach of [4] we avoid encoding the complete complex in a traditional mesh hierarchy. In-stead, we encode a reduced complex created by disregarding some topological constraints on the complex. The corresponding data is stored separately in a structure called cancellation forest. Concep-tually, a cancellation forest consists of sets of critical points gov-erned by the concepts of Morse theory. The combination of this new structure with a traditional mesh hierarchy proofs to be signif-icantly more flexible than the one previously reported [4]. In par-ticular, the resulting hierarchy is guaranteed to be of logarithmic height. 1