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.
The application of the interleaving distance to labeled merge trees can also be viewed as an interpretation of a metric for phylogenetic trees @cite_68 . Our framework differs from the previous work as it relies on a clean and simple metric-space view of input trees that is equipped with geodesics, as well as easy-to-implement algorithms.
@cite_68
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Background Phylogenetic tree comparison metrics are an important tool in the study of evolution, and hence the definition of such metrics is an interesting problem in phylogenetics. In a paper in Taxon fifty years ago, Sokal and Rohlf proposed to measure quantitatively the difference between a pair of phylogenetic trees by first encoding them by means of their half-matrices of cophenetic values, and then comparing these matrices. This idea has been used several times since then to define dissimilarity measures between phylogenetic trees but, to our knowledge, no proper metric on weighted phylogenetic trees with nested taxa based on this idea has been formally defined and studied yet. Actually, the cophenetic values of pairs of different taxa alone are not enough to single out phylogenetic trees with weighted arcs or nested taxa.