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.
Graph-based topological descriptors. Graph-based topological descriptors include merge trees @cite_20 (also known as barrier trees @cite_37 or join trees @cite_17 ), contour trees @cite_17 , Reeb graphs @cite_16 , mapper graphs @cite_12 , and joint contour nets @cite_27 . These descriptors are graph-based representations to illustrate how the topology of level sets or sublevel sets of scalar fields changes with a scalar value parameter.
@cite_37 @cite_27 @cite_16 @cite_20 @cite_12 @cite_17
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The heights of energy barriers separating two (macro-)states are useful for estimating transition frequencies. In non-degenerate landscapes the decomposition of a landscape into basins surrounding local minima connected by saddle points is straightforward and yields a useful definition of macro-states. In this work we develop a rigorous concept of barrier trees for degenerate landscapes. We present a program that efficiently computes such barrier trees, and apply it to two well known examples of landscapes. Contour trees and Reeb graphs are firmly embedded in scientific visualization for analysing univariate (scalar) fields. We generalize this analysis to multivariate fields with a data structure called the Joint Contour Net that quantizes the variation of multiple variables simultaneously. We report the first algorithm for constructing the Joint Contour Net and demonstrate that Contour Trees for individual variables can be extracted from the Joint Contour Net. We present a computational method for extracting simple descriptions of high dimensional data sets in the form of simplicial complexes. Our method, called Mapper, is based on the idea of partial clustering of the data guided by a set of functions defined on the data. The proposed method is not dependent on any particular clustering algorithm, i.e. any clustering algorithm may be used with Mapper. We implement this method and present a few sample applications in which simple descriptions of the data present important information about its structure. We show that contour trees can be computed in all dimensions by a simple algorithm that merges two trees. Our algorithm extends, simplifies, and improves work of Tarasov and Vyalyi and of van