GVU Technical Report Number: GIT-GVU-05-09

Title: Simplifying the Topology of Volume Datasets: An Opportunistic Approach

Authors: Andrzej Szymczak, James Vanderhyde

Abstract:
Understanding isosurfaces and contours (their connected components) is important for the analysis as well as effective visualization of 3D scalar fields. The topological changes that the contours undergo as the isovalue varies are typically represented using the contour tree, which can be obtained from the input scalar field by collapsing every contour to a single point. Contour trees are known to provide useful information, allowing one to find interesting isovalues and contours, speed up computations involving isosurfaces or contours, or analyze or visualize the scalar field's qualitative structure. However, the applicability of contour trees can, in many cases, be problematic because of their large size.

Morse theory relates the contour topology changes to critical points in the underlying scalar fields. We describe a simple algorithm that can decrease the number of critical points in a regularly sampled volume dataset. The procedure produces a perturbed version of the input volume that has fewer critical points but, at the same time, is guaranteed to be less than a user-specified threshold away from the input volume (in the supremum norm sense). Because the input and output volumes are close, the algorithm preserves the most stable topological features of the scalar field. Although we do not guarantee that the number of critical points in the output volume is minimum among all volumes within the threshold away from the input dataset, our experiments demonstrate that the procedure is quite effective for a variety of input data types. Apart from reducing the size of the contour tree, it also reduces the topological complexity of individual isosurfaces.

Keywords: Isosurface, topology, genus

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