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Efficient Computation of a Hierarchy of Discrete 3D Gradient Vector Fields


Weinkauf,  Tino
Computer Graphics, MPI for Informatics, Max Planck Society;

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Günther, D., Reininghaus, J., Prohaska, S., Weinkauf, T., & Hege, H.-C. (2012). Efficient Computation of a Hierarchy of Discrete 3D Gradient Vector Fields. In R. Peikert, H. Hauser, H. Car, & R. Fuchs (Eds.), Topological Methods in Data Analysis and Visualization II (pp. 15-29). New York, NY: Springer.

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.