We have shown two approaches to a Morse decomposition in three dimensions and different possibilities to classify the obtained Morse sets. We found out in our experiments, that for a large fixed integration length, there are still Morse sets remaining, that cannot be further decomposed, so there is plenty of potential in improving the streamline-based algorithm, i.e., repeatedly applying it to the remaining sets with increasing arc-length parameter. In addition, a parallelization must be considered as a must in future work. The main difference to classical topology is, that the equivalence classes of streamlines are not induced by having the same!- and˛- limit set by integration, but being in the same strongly connected component. Though the complex shape of Morse sets has theoretically a higher variation in three dimensions, practically fixed points of saddle character will dominate in data obtained from CFD simulations. The clusters of cells do not always give an immediate insight into the behaviour of the field, but it still can be used as a preprocessing algorithm for finer techniques. An interesting challenge in the future is to make general conclusions in how exactly the size of the grid and the length of integration will influence the results.
Acknowledgements We want to thank Tomasz Kaczynski, Matthias Schwarz, the people from the Krakau research group for “Computer Assisted Proofs in Dynamics” and the reviewers for many valuable hints and comments. We also thank the DFG for funding the project by grant SCHE 663/3-8.
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Statistics
Gunther H. Weber, Peer-Timo Bremer, and Valerio Pascucci
1 Introduction
Isosurfaces [10,13,14] are an extremely versatile and ubiquitous component of data analysis since distinct isosurfaces often have a useful physical interpretation directly related to the application domain. In premixed combustion simulations, for example, an isotherm (isosurfaces of temperature) of the appropriate temperature is often associated with the location of the flame; in molecular simulations, isosurfaces of a certain charge density indicate molecular boundaries; and in fluid simulations, isosurfaces can be used to identify an interface between mixing fluids. Furthermore, isosurfaces provide a geometric representation of scientific data that supports the definition of further derived quantities (such as surface area) that are useful for an in-depth analysis.
Due to their importance, isosurfaces and in particular their topology have been the focus of an extensive body of research. In particular, the dependence of isosurface properties as a function of the isovalue has been examined in detail.
Traditionally, there are two complementary types of information derived to aid in identifying interesting and relevant isosurfaces. Structural approaches, often based
G.H. Weber ()
Computational Research Division, Lawrence Berkeley National Laboratory, One Cyclotron Road, Berkeley, CA 94720, USA
e-mail:GHWeber@lbl.gov P.-T. Bremer
Center of Applied Scientific Computing, Lawrence Livermore National Laboratory, Box 808, L-560, Livermore, CA 94551, USA
e-mail:bremer5@llnl.gov V. Pascucci
Scientific Computing and Imaging Institute, University of Utah, 72 South Central Campus Drive, Salt Lake City, UT 84112, USA
e-mail:pascucci@sci.utah.edu
R. Peikert et al. (eds.), Topological Methods in Data Analysis and Visualization II, Mathematics and Visualization, DOI 10.1007/978-3-642-23175-9 5,
© Springer-Verlag Berlin Heidelberg 2012
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on the contour tree, provide information about individual contours, the connected components of isosurfaces, and how their number changes as the isovalue varies.
The other class of approaches, including the contour spectrum, derive quantitative measurements, such as surface area from the isosurfaces, and plot these quantities as a function of isovalue. While providing quantitative information about the isosurface, information about individual connected components is lost, and these approaches usually result in a single one-dimensional function plot.
It is possible to combine these two complementary techniques by considering the segmentation implied by the contour tree. Each arc in the contour tree corresponds to a region swept by the contours represented by the arc. Using this segmentation, it is possible to calculate measurements on a per-contour basis. In this paper, we introducetopological cacti, a new visualization metaphor for the contour tree that incorporates these segmentation-based quantitative measurements into the graph display. Like the toporrery [15], topological cacti utilize a radial graph layout in three dimensions to avoid self-intersections. However, topological cacti also support the display of quantitative measures along with the graph. We draw graph edges of the toporerry as cylindrical extrusions and map a selected quantity to graph edge radius. We augment this visualization by adding spikes to display a second quantity, thus completing the cactus metaphor.
We will now describe how to display topological cacti. As a first step, we calculate measurements based on the segmentation implied by the contour tree (Sect.3). In particular, we compute the contour spectrum for individual contours to provide per-contour area and volume measures (Sect.3.1). Subsequently, we describe how to use this information in layout computation of topological cacti (Sect.4). We illustrate the concept of topological cacti on example data sets using the per-contour representation of the contour spectrum (Sect.5). We also demonstrate the display of other derived quantities, such as those associated with a join/merge tree resulting from the analysis of combustion simulations, as well as merge trees derived from higher-dimensional data.