A Deformation-based Edit Distance for Merge Trees

TL;DR

This paper introduces a novel deformation-based edit distance for merge trees in scientific visualization, enabling more accurate and topology-aware comparisons of scalar fields with a new algorithm that is both efficient and metric.

Contribution

It proposes a new set of deformation operations directly on merge trees, differing from traditional branch decomposition methods, and provides a quartic time algorithm for this metric.

Findings

The new edit distance is branch decomposition-independent.

It is a metric on the set of all merge trees.

The algorithm runs in quartic time.

Abstract

In scientific visualization, scalar fields are often compared through edit distances between their merge trees. Typical tasks include ensemble analysis, feature tracking and symmetry or periodicity detection. Tree edit distances represent how one tree can be transformed into another through a sequence of simple edit operations: relabeling, insertion and deletion of nodes. In this paper, we present a new set of edit operations working directly on the merge tree as an geometrical or topological object: the represented operations are deformation retractions and inverse transformations on merge trees, which stands in contrast to other methods working on branch decomposition trees. We present a quartic time algorithm for the new edit distance, which is branch decomposition-independent and a metric on the set of all merge trees.