A Comparative Study of Different Edit Distance-Based Methods for Feature Tracking using Merge Trees on Time-Varying Scalar Fields

TL;DR

The study addresses feature tracking in time-varying scalar fields by comparing four merge-tree edit-distance methods: ${d_e}$, ${d_w}$, ${d_b}$, and ${d_p}$. It analyzes their theoretical bases, stability to instabilities and noise, and performance on synthetic Cylinder2D and real HeartBeat3D data, including pairwise similarities. Key findings show substantial differences across methods and datasets, with branch-decomposition-independent approaches generally offering more stable tracking under small perturbations, while branching-focused methods can be more sensitive to instabilities; a look-ahead in ${d_p}$ improves robustness. The work provides practical guidance for selecting tracking methods, highlights limitations in current evaluation practices, and calls for standardized evaluation frameworks for merge-tree based tracking.

Abstract

Feature tracking in time-varying scalar fields is a fundamental task in scientific computing. Topological descriptors, which summarize important features of data, have proved to be viable tools to facilitate this task. The merge tree is a topological descriptor that captures the connectivity behaviors of the sub- or superlevel sets of a scalar field. Edit distances between merge trees play a vital role in effective temporal data tracking. Existing methods to compute them fall into two main classes, namely whether they are dependent or independent of the branch decomposition. These two classes represent the most prominent approaches for producing tracking results. In this paper, we compare four different merge tree edit distance-based methods for feature tracking. We demonstrate that these methods yield distinct results with both analytical and real-world data sets. Furthermore, we investigate how these results vary and identify the factors that influence them. Our experiments reveal significant differences in tracked features over time, even among those produced by techniques within the same category.

Figures (10)

• Figure 1: Pressure field of two time steps of HeartBeat3D.
• Figure 2: Shown are two types of instability. In the left figure, nodes $A$ and $B$ change their values, causing the other to become the new global maximum, which modifies the BDT. Methods that rely on BDTs struggle with this type of instability. The right figure illustrates the horizontal instability. Saddle nodes $E$ and $F$ have very close values. A small perturbation can swap their positions and consequently change the structure of the tree. This type of instability affect most methods.
• Figure 3: Similar features determined by the chosen methods between two very similar scalar fields. The fields differ slightly, which trigger the instabilities between two merge trees. There is no difference between the results of the constrained edit distance sridharamurthy2018edit, branch mapping distance wetzels2022branch and path mapping distance wetzels2022path. All chosen methods, even for the path mapping distance wetzels2025stable with a high look-ahead, are affected by these instabilities and fail to produce the expected result, that is all horizontal lines should be parallel.
• Figure 4: Noisy scalar field created from a cleaned scalar field using our strategy. The noises are added such that the local minima in the merge tree of the noisy field includes the local minima of the cleaned field.
• Figure 5: The tracking stability of the extrema of four methods If we stop tracking at $\epsilon_\text{max} = 5\%$. The median, first and third quantiles, maximum and minimum values of the stability scores at each level of affected points are shown for each method. The stability score decreases as the percentage of affected points and the level of noise increase. In general, $d_w$ produces the tracking with the lowest scores, while the other three methods behave quite similarly. The path mapping distance $d_p$ with look-ahead of $4$ yields slightly better and more stable scores compared to $d_e$ and $d_p$.