Analyzing Multifaceted Scalar fields through Topological Similarity
Reeb graphs have been used in a number of applications including analyzing the topological properties of unifaceted scalar fields. It is natural to consider the situation of analyzing multiple scalar fields arising from multifaceted data via Reeb graphs. Several distances have been defined on Reeb graphs, three of which we will focus on here: The interleaving distance, the functional distortion distance, and the Reeb graph edit distance. Each distance pulls insipiration from different mathematical fields such as category theory, metric and Banach spaces, and sequence and string matching. Here, we provide an abridged construction for each distance as well as an analysis of the distance properties that each exhibit. We then discuss future directions for analysis of scalar fields using Reeb graph metrics either through these aforementioned distances or by taking a machine learning approach for graph similarity learning.