Abstract
This paper introduces a novel combinatorial algorithm to compute a hierarchy of discrete gradient vector fields for three-dimensional scalar fields. The hierarchy is defined by an importance measure and represents the combinatorial gradient flow at different levels of detail. The presented algorithm is based on Forman’s discrete Morse theory, which guarantees topological consistency and algorithmic robustness. In contrast to previous work, our algorithm combines memory and runtime efficiency. It thereby lends itself to the analysis of large data sets. A discrete gradient vector field is also a compact representation of the underlying extremal structures – the critical points, separation lines and surfaces. Given a certain level of detail, an explicit geometric representation of these structures can be extracted using simple and fast graph algorithms.
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Acknowledgements
This work was supported by the Max-Planck Institute of Biochemistry, Martinsried, and the DFG Emmy-Noether research program. The authors would like to thank Daniel Baum, Ingrid Hotz, Jens Kasten, Michael Koppitz, Falko Marquardt, and Jan Sahner for many fruitful discussions on this topic.
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Günther, D., Reininghaus, J., Prohaska, S., Weinkauf, T., Hege, HC. (2012). Efficient Computation of a Hierarchy of Discrete 3D Gradient Vector Fields. In: Peikert, R., Hauser, H., Carr, H., Fuchs, R. (eds) Topological Methods in Data Analysis and Visualization II. Mathematics and Visualization. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-23175-9_2
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