Comparative Analysis of Merge Trees using Local Tree Edit Distance

TL;DR

Local topology analysis requires substructure-aware comparisons of merge trees. The paper introduces the local tree edit distance $lmted$, a metric-based local variant of the merge-tree distance, and develops a dynamic-programming algorithm with a truncated-persistence cost to enable efficient, multi-scale subtree matching. It proves metric properties, provides refinements and optimizations, and demonstrates applications in symmetry detection, subsampling/smoothing analysis, topology-preserving compression, and spatio-temporal feature tracking. The approach offers fine-grained, local insights beyond global measures, with potential impact on visualization, analysis, and time-varying/ensemble data studies.

Abstract

Comparative analysis of scalar fields is an important problem with various applications including feature-directed visualization and feature tracking in time-varying data. Comparing topological structures that are abstract and succinct representations of the scalar fields lead to faster and meaningful comparison. While there are many distance or similarity measures to compare topological structures in a global context, there are no known measures for comparing topological structures locally. While the global measures have many applications, they do not directly lend themselves to fine-grained analysis across multiple scales. We define a local variant of the tree edit distance and apply it towards local comparative analysis of merge trees with support for finer analysis. We also present experimental results on time-varying scalar fields, 3D cryo-electron microscopy data, and other synthetic data sets to show the utility of this approach in applications like symmetry detection and feature tracking.

Figures (19)

• Figure 1: Merge trees. (a) A 2D scalar field together with its critical points and a set of isocontours. (b,c) A merge tree tracks the connectivity of sublevel sets (preimage $f^{-1} (-\infty, c]$) or the superlevel sets (preimage $f^{-1} [c,\infty)$). We consistently use the () color map for the scalar field and the () color map for representing critical points based on their Morse index (0: minimum, 1: 1-saddle, 2: 2-saddle, 3: maximum).
• Figure 2: Scalar fields $f_1$ and $f_2$ that are not globally similar but contain locally similar regions. Split trees are overlaid on top of the scalar fields
• Figure 3: Split trees corresponding to the scalar fields shown in Figure \ref{['fig:local']}. One persistence pair is shown within each split tree (orange link). When comparing subtrees $T_1[i_8]$ or $T_2[j_5]$, nodes $i_5$ and $j_4$ are unpaired. So, dummy nodes $i'_{10}$ and $j'_7$ are inserted with $|f_1(i_{10}) - f_1(i'_{10})| < \varepsilon$ and $|f_2(j_7) - f_2(j'_7)| < \varepsilon$ to serve as root and as a pair of the unpaired node.
• Figure 4: Illustrating lmted. To compute lmted between subtrees rooted at $i$ and $j$, we treat the subtrees containing the unpaired nodes $i_u,j_u$, labeled as $i_{u_i},j_{u_j}$ and highlighted in orange, differently. For other nodes, the formulation is same as mted. For $i_u,j_u$, we use truncated persistence to determine the costs. In the matching required to compute $M'_r(i,j)$, we consider truncated persistence to determine the weights of all edges incident on $i_{u_i},j_{u_j}$ (highlighted in orange). $s$ and $t$ are the source and destination nodes of the flow problem that is equivalent to the matching problem to determine $M'_r(i,j)$. The cardinalities of the two sides are made equal by inserting a set of $n_j$ dummy nodes adjacent to $s$ and $n_i$ dummy nodes adjacent to $t$.
• Figure 5: Illustration of Cases O and M. The portion of the path colored in red corresponds to case M while the portion of the path colored in green corresponds to case O.