Topological Methods in Data Analysis and Visualization III by Peer-Timo Bremer Ingrid Hotz Valerio Pascucci & Ronald Peikert
Author:Peer-Timo Bremer, Ingrid Hotz, Valerio Pascucci & Ronald Peikert
Language: eng
Format: epub
Publisher: Springer International Publishing, Cham
The motivation behind these two simplification strategies is different. While the explicit procedure refers to an explicit computation of a true simplified scalar field, the implicit case directly addresses the (combinatorial) gradient flow. However, both simplification procedures follow the same rules to determine which simplification step is admissible and they may also follow both the same principle to determine the next removal. But the results of the explicit and implicit simplification procedures differ, in general.
For 2D data sets, the difference is limited to the geometric embedding of the separation lines. For 3D data sets, on the other hand, the simplification procedures may even lead to different hierarchies: the order of the removal as well as the connectivity of critical points might differ. In this work, we investigate specific properties of the two representations and show that these properties can cause these differences. The different representations also affect the algorithmic side. Therefore, we also provide a complexity analysis of the two simplification schemes.
In Sect. 2, we provide a brief overview of prior work. Both simplification strategies are presented in Sect. 3, and we discuss specific properties in Sect. 4. We conclude this chapter in Sect. 5 by discussing the effects of these properties.
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