Edit Distance between Merge Trees

TL;DR

The paper introduces a metric tree edit distance between merge trees to compare scalar fields, enabling feature-directed analysis of time-varying data. It develops a merge-tree–tailored cost model with two concrete edit costs and proves metric properties, including stability under perturbations. A stabilization strategy addresses instabilities, and the authors provide a dynamic-programming algorithm suitable for practical computation. Extensive experiments across time-varying data, symmetry detection, shape matching, and data summarization demonstrate that the distance captures meaningful topological differences beyond traditional persistence-based measures. The approach offers a principled, topology-aware tool for analyzing scalar fields in scientific visualization and related applications.

Abstract

Topological structures such as the merge tree provide an abstract and succinct representation of scalar fields. They facilitate effective visualization and interactive exploration of feature-rich data. A merge tree captures the topology of sub-level and super-level sets in a scalar field. Estimating the similarity between merge trees is an important problem with applications to feature-directed visualization of time-varying data. We present an approach based on tree edit distance to compare merge trees. The comparison measure satisfies metric properties, it can be computed efficiently, and the cost model for the edit operations is both intuitive and captures well-known properties of merge trees. Experimental results on time-varying scalar fields, 3D cryo electron microscopy data, shape data, and various synthetic datasets show the utility of the edit distance towards a feature-driven analysis of scalar fields.

Figures (20)

• Figure 1: Merge trees. (a) A 2D scalar field (b) A merge tree tracks the connectivity of sub-level sets (preimage of $f^{-1} (-\infty, c]$) or the super-level sets (preimage of $f^{-1} [c,\infty)$).
• Figure 2: Persistence pairs in the join (left) and split (right) trees.
• Figure 3: A 1D scalar function (left) and the persistence diagram of the function (right). Each birth-death pair $(b_i,d_i)$ is a feature of the scalar function and its persistence is defined as $d_i - b_i$. Each pair is represented as a point in $\mathbb{R}^2$.
• Figure 4: The discriminative power of the bottleneck distance $D_B$ is low. Two scalar functions (blue and red) and the corresponding persistence diagrams and merge trees. Even though the scalar functions are different, $D_B$ is not able to capture the difference because the persistence diagrams are equal. A distance measure that considers the structure of the merge tree would discriminate the two scalar functions.
• Figure 5: Three different tree edit operations. Each edit affects only one node in the tree. The null character $\lambda$ corresponds to a gap.