Physical phenomena in science and engineering are frequently modeled using scalar fields. In scalar field topology, graph-based topological descriptors such as merge trees, contour trees, and Reeb graphs are commonly used to characterize topological changes in the (sub)level sets of scalar fields. One of the biggest challenges and opportunities to advance topology-based visualization is to understand and incorporate uncertainty into such topological descriptors to effectively reason about their underlying data. In this paper, we study a structural average of a set of labeled merge trees and use it to encode uncertainty in data. Specifically, we compute a 1-center tree that minimizes its maximum distance to any other tree in the set under a well-defined metric called the interleaving distance. We provide heuristic strategies that compute structural averages of merge trees whose labels do not fully agree. We further provide an interactive visualization system that resembles a numerical calculator that takes as input a set of merge trees and outputs a tree as their structural average. We also highlight structural similarities between the input and the average and incorporate uncertainty information for visual exploration. We develop a novel measure of uncertainty, referred to as consistency, via a metric-space view of the input trees. Finally, we demonstrate an application of our framework through merge trees that arise from ensembles of scalar fields. Our work is the first to employ interleaving distances and consistency to study a global, mathematically rigorous, structural average of merge trees in the context of uncertainty visualization.
Uncertainty is a common and crucial issue in scientific data. The exploration and analysis of three-dimensional (3D) and large two-dimensional (2D) data with uncertainty information demand an effective visualization augmented with both user interaction and relevant context. The contour tree has been exploited as an efficient data structure to guide exploratory visualization. This paper proposes an interactive visualization tool for exploring data with quantitative uncertainty representations. First, we introduce a balanced planar hierarchical contour tree layout integrated with tree view interaction, allowing users to quickly navigate between levels of detail for contours of large data. Further, uncertainty information is attached to a planar contour tree layout to avoid the visual cluttering and occlusion in viewing uncertainty in 3D data or large 2D data. For the first time, the uncertainty information is explored as a combination of the data-level uncertainty which represents the uncertainty concerning the numerical values of the data, the contour variability which quantifies the positional variation of contours, and the topology variability which reveals the topological variation of contour trees. This information provides a new insight into how the uncertainty exists with and relates to the features of the data. The experimental results show that this new visualization facilitates a quick and accurate selection of prominent contours with high or low uncertainty and variability. We study contour trees of terrains, which encode the topological changes of the level set of the height value e as we raise e from -∞ to +∞ on the terrains, in the presence of uncertainty in data. We assume that the terrain is represented by a piecewise-linear height function over a planar triangulation M, by specifying the height of each vertex. We study the case when M is fixed and the uncertainty lies in the height of each vertex in the triangulation, which is described by a probability distribution. We present efficient sampling-based Monte Carlo methods for estimating, with high probability, (i) the probability that two points lie on the same edge of the contour tree, within additive error; (ii) the expected distance of two points p, q and the probability that the distance of p, q is at least e on the contour tree, within additive error, where the distance of p, q on a contour tree is defined to be the difference between the maximum height and the minimum height on the unique path from p to q on the contour tree. The main technical contribution of the paper is to prove that a small number of samples are sufficient to estimate these quantities. We present two applications of these algorithms, and also some experimental results to demonstrate the effectiveness of our approach. Contour trees can represent the topology of large volume data sets in a relatively compact, discrete data structure. However, the resulting trees often contain many thousands of nodes; thus, many graph drawing techniques fail to produce satisfactory results. Therefore, several visualization methods were proposed recently for the visualization of contour trees. Unfortunately, none of these techniques is able to handle uncertain contour trees although any uncertainty of the volume data inevitably results in partially uncertain contour trees. In this work, we visualize uncertain contour trees by combining the contour trees of two morphologically filtered versions of a volume data set, which represent the range of uncertainty. These two contour trees are combined and visualized within a single image such that a range of potential contour trees is represented by the resulting visualization. Thus, potentially erroneous topological structures are visually distinguished from more certain structures. Moreover, topological structures can be revealed that are otherwise obscured by data errors. We present and discuss results obtained with a prototypical implementation using well-known volume data sets. Most visualization systems fail to convey uncertainty within data. To provide a way to show uncertainty in similar hierarchies, we interpreted the differences between two tree structures as uncertainty. We developed a new interactive visualization system called CandidTree that merges two trees into one and visualizes two types of structural uncertainty: location and sub-tree structure uncertainty. Since CandidTree can visualize the differences between two tree structures, we conducted a series of user studies with tree-comparison tasks. First a usability study was conducted to identify major usability issues and evaluate how our system works. Another qualitative user study was conducted to see if biologists, who regularly work with hierarchically organized names, are able to use CandidTree, and to assess the 'uncertainty' metric we used. A controlled experiment with software engineers was conducted to compare CandidTree with WinDiff, a traditional files and folders comparison tool. The results showed that users performed better with CandidTree. Furthermore, CandidTree received better satisfaction ratings and all users preferred CandidTree to WinDiff.