  Merge trees, a type of topological descriptor, serve to identify and
summarize the topological characteristics associated with scalar fields. They
present a great potential for the analysis and visualization of time-varying
data. First, they give compressed and topology-preserving representations of
data instances. Second, their comparisons provide a basis for studying the
relations among data instances, such as their distributions, clusters,
outliers, and periodicities. A number of comparative measures have been
developed for merge trees. However, these measures are often computationally
expensive since they implicitly consider all possible correspondences between
critical points of the merge trees. In this paper, we perform geometry-aware
comparisons of merge trees using labeled interleaving distances. The main idea
is to decouple the computation of a comparative measure into two steps: a
labeling step that generates a correspondence between the critical points of
two merge trees, and a comparison step that computes distances between a pair
of labeled merge trees by encoding them as matrices. We show that our approach
is general, computationally efficient, and practically useful. Our general
framework makes it possible to integrate geometric information of the data
domain in the labeling process. At the same time, it reduces the computational
complexity since not all possible correspondences have to be considered. We
demonstrate via experiments that such geometry-aware merge tree comparisons
help to detect transitions, clusters, and periodicities of time-varying
datasets, as well as to diagnose and highlight the topological changes between
adjacent data instances.
