Topological Cacti: Combining Structural and Quantitative Information
Problem Statement and Goals
Current visualization techniques provide either structural or quantitative information based on topological analysis over a data set. Contours, the connected components of level sets, play an important role in understanding the global structure of a scalar field. In particular, their nesting behavior and topology—often represented in the form of a contour tree—has been used extensively for visualization and analysis. Traditional contour tree visualizations only encode structural properties like the number of contours or the nesting of contours, but no quantitative information such as volume or other statistics. To gain full understanding of data sets—for example, to gain insights in the size distribution of individual connected isosurface compinents—it is desirable to combine these complementary views into a single visualization.
We introduced a new visual metaphor for contour trees, called topological cacti , that extends the traditional toporrery display of a contour tree to display additional quantitative information as the width of the cactus trunk and the length of its spikes. To demonstrate the effectiveness of this approach, we have applied this new technique to scalar fields with varying dimensions and different measures.
Implementation and Results
The contour tree captures the topological evolution of an isosurface as the isovalue varies. Critical points, where the number of contours changes, appear as nodes in the contour tree. Nodes of degree one (leaves of the tree) correspond to extrema, where contours are created or destroyed. Interior nodes of degree three or higher correspond to "saddles," where two or more contours merge or split. Arcs of the contour tree represent contours between critical points, i.e., contours that do not change topology (with the exception of genus changes) as the isovalue varies between critical values.
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. We introduce topological cacti, a new visualization metaphor for the contour tree, which incorporates these segmentation-based quantitative measurements into the graph display.
We use a radial graph layout for generating topological cacti. Extrema are ordered in concentric discs, where the disc is chosen based on the depth of an extremum in the branch decomposition. To display metrics, we draw each branch as cylindrical extrusion and chose the radii according to a first selected measure (clamping, however, at a minimum radius to ensure that branches do not disappear). Before mapping a metric such as volume to radius, we support choosing a variety of transformation functions (square root, cube root, logarithm) to ensure that higher dimensional properties are appropriately scaled to the one-dimensional radius.
For the display of a second metric, we augment the cylindrical extrusions with spikes. We start by placing circles along regular intervals of the cylindrical extrusion and choosing angles in regular intervals along this circle. We then choose spike location by jittering these regular samples, i.e., we add random offset angles and heights to the originally chosen spike locations. Spikes are drawn as cones with a fixed (user defined) base radius and a length depending on the metrics. Similar to the extrusion radius, we support user chosen mappings such as square root, cube root or logarithm.
Figure 1 shows an example of a topological cactus for the hydrogen atom data set (simplified to a persistence of ten). This is a simple, small—1283 byte sample—example data set that we use to illustrate the concept. We display the volume (mapped via cube root) as branch width and the surface area (square root) as spike length, to examine whether persistent features also correspond to large features. The root branch corresponds to the evolution of the contour in the center. It is immediately obvious that despite its very high persistence (resulting in it being chosen as the root branch), both its volume and surface area are very small. The two lobes, corresponding to the two symmetric branches are much larger in both volume and surface area. Finally, the ring, corresponding to the lowest branch, has the largest volume and area.
Figure 2 shows the topological cactus of the merge tree of fuel consumption rate in a combustion simulation (one 4.3GByte timestep from a coarse AMR simulation). Here, we are interested in how the size of the burning regions correlates with temperature variance. Thus, we map the cube root of the volume—approximated as the number of vertices—to branch width and temperature variance to spike length.
The cactus shows that there are two types of regions: (i) branches that are relatively wide close to the maximum (rectangle "A" in the figure marks an example), corresponding to regions comprised of a relatively large number of vertices and (ii) branches that are thin close to the maximum (rectangle "B" in the figure marks an example), corresponding to small regions. The latter occur at lower fuel-consumption levels (i.e., their maxima are generally lower than those of the wide branches). The spikes show that temperature variance is small where there is high fuel consumption, which occurs where there is "intense burning" (where fuel consumption is high). Furthermore, the temperature variance is expectedly larger for non-burning regions (regions of no to low fuel consumption).
Figure 3 shows applying our method in the context of uncertainty quantification. The cactus shows the merge tree of a 21-dimensional climate simulation ensemble. The data set consists of 1197 climate simulations performed with the Community Atmospheric Model (CAM)—each using a different combination of 21 input parameters. Such ensembles are used to study the sensitivity of the climate predictions to changes in input parameters. The output function shown in Figure 3 s the average longwave energy output (FLUT) in the winter months (December, January, February). The cactus shows that most of the "volume" of the parameter space is collected in several large but relatively low persistent branches of maxima. This fact is interesting as the observed energy output is expected to be closer to the global minimum. Thus, it appears that only a small portion of the input samples produces realistic outputs. Nevertheless, all persistences are fairy low indicating that the overall behavior is stable the under perturbation of input parameters.
Topological cacti combine insights obtained from the contour tree with metrics derived on a per contour bases. This combination can greatly enhance the expressiveness of contour tree representations, in particular when the importance measured by persistence differs from other metrics, such as area of volume. The contour spectrum implementation that associates the contour spectrum with the contour tree also has applications in other topological data analysis settings, such as contour tree simplification based on these metrics or deriving quantitative measurements from the contour tree.
Peer-Timo Bremer (LLNL), Valerio Pascucci (University of Utah).