Branch Decomposition-Independent Edit Distances for Merge Trees

TL;DR

This paper introduces branch mappings for merge tree edit distances, providing a more stable, decomposition-independent method that improves computational efficiency and practical applicability in scientific visualization tasks.

Contribution

It proposes a novel branch mapping approach that is independent of branch decompositions, enhancing stability and efficiency in merge tree comparison.

Findings

Faster quartic runtime algorithm for branch mappings

More stable distance measure for topological features

Effective on synthetic and real-world data

Abstract

Edit distances between merge trees of scalar fields have many applications in scientific visualization, such as ensemble analysis, feature tracking or symmetry detection. In this paper, we propose branch mappings, a novel approach to the construction of edit mappings for merge trees. Classic edit mappings match nodes or edges of two trees onto each other, and therefore have to either rely on branch decompositions of both trees or have to use auxiliary node properties to determine a matching. In contrast, branch mappings employ branch properties instead of node similarity information, and are independent of predetermined branch decompositions. Especially for topological features, which are typically based on branch properties, this allows a more intuitive distance measure which is also less susceptible to instabilities from small-scale perturbations. We describe a quartic runtime algorithm for computing optimal branch mappings, which is faster than the only other branch decomposition-independent method in the literature by more than a linear factor. Furthermore, we compare the results of our method on synthetic and real-world examples to demonstrate its practicality and utility.