Maximizing adaptivity in hierarchical topological models

P.-T. Bremer, V. Pascucci, B. Hamann
International Conference on Shape Modeling and Applications 2005 (SMI' 05)  
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 produce a highly adaptive topology-based
more » ... ution representation of a given function. Contrary to the approach of [4] we avoid encoding the complete complex in a traditional mesh hierarchy. Instead, 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. Conceptually, a cancellation forest consists of sets of critical points governed by the concepts of Morse theory. The combination of this new structure with a traditional mesh hierarchy proofs to be significantly more flexible than the one previously reported [4] . In particular, the resulting hierarchy is guaranteed to be of logarithmic height. Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. To copy otherwise, to republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Copyright 200X ACM X-XXXXX-XX-X/XX/XX ...$5.00. adaptive in order to be useful for a large variety of situations. Requiring interactivity inadvertently leads to the use of hierarchical encodings rather than simplification schemes. Hierarchical models often reduce the adaptivity of a representation to gain the ability to preform incremental changes for varying queries. We address the need for adaptive topology-based data exploration by improving significantly the topological hierarchy described in [4] . By encoding a less specific complex as multi-resolution mesh and the remaining information separately as a collection of sets, we show how one can remove many of the dependencies in the original hierarchy, making the structure simpler, more compact, and more adaptive than the original one.
doi:10.1109/smi.2005.28 dblp:conf/smi/BremerPH05 fatcat:2conadsej5a43bcyfuya7mvikq