ParkView: Visualizing Monotone Interleavings

TL;DR

ParkView tackles visualizing monotone interleavings of ordered merge trees by introducing a two-level decomposition into path and hedge structures. It proves that heavy path-branch decompositions minimize visual complexity and that hedges are 3-colorable, enabling a concise, color-coded display of both interleaving maps. The method yields a linear-time algorithm and scalable visuals demonstrated on real-world datasets, with grid lines and active paths conveying the interleaving height $\delta$ and correspondences. While effective for two trees, extending to multi-tree comparisons and interactive exploration remains future work, along with refining the interleaving distance to capture local similarity.

Abstract

Merge trees are a powerful tool from topological data analysis that is frequently used to analyze scalar fields. The similarity between two merge trees can be captured by an interleaving: a pair of maps between the trees that jointly preserve ancestor relations in the trees. Interleavings can have a complex structure; visualizing them requires a sense of (drawing) order which is not inherent in this purely topological concept. However, in practice it is often desirable to introduce additional geometric constraints, which leads to variants such as labeled or monotone interleavings. Monotone interleavings respect a given order on the leaves of the merge trees and hence have the potential to be visualized in a clear and comprehensive manner. In this paper, we introduce ParkView: a schematic, scalable encoding for monotone interleavings. ParkView captures both maps of the interleaving using an optimal decomposition of both trees into paths and corresponding branches. We prove several structural properties of monotone interleavings, which support a sparse visual encoding using active paths and hedges that can be linked using a maximum of 6 colors for merge trees of arbitrary size. We show how to compute an optimal path-branch decomposition in linear time and illustrate ParkView on a number of real-world datasets.

Figures (15)

• Figure 1: A 2D scalar field with its minima and saddle points marked (left) and the corresponding merge tree (right).
• Figure 1: ParkView of a monotone interleaving using six colors of the same hue, with different lightness values, per tree. The merge trees (73 leaves left, 71 leaves right) are derived from Timesteps 177 (left) and 178 (right) of the Ionization Front dataset pont2022wasserstein, using a persistence simplification threshold of 0.01. Grid lines at distance of $\delta$.
• Figure 2: Two $\delta$-shift maps $\alpha$ and $\beta$ that define a monotone interleaving between ordered merge trees $T$ and $T'$. Note that $u_1 \sqsubseteq u_2 \sqsubseteq u_3$ and $v_1 \sqsubseteq' v_2$. The compositions $\alpha \circ \beta$ and $\beta \circ \alpha$ map $u_3$ and $v_2$ to their respective ancestor at height $2\delta$ higher, grid lines at distance $\delta$.
• Figure 2: ParkView of a monotone interleaving using six colors, that have distinct hues, per tree. The merge trees (73 leaves left, 71 leaves right) are derived from Timesteps 177 (left) and 178 (right) of the Ionization Front dataset pont2022wasserstein, using a persistence simplification threshold of 0.01. Grid lines at distance of $\delta$.
• Figure 3: Visualizing interleavings.

Theorems & Definitions (18)

• Lemma 1
• proof
• Theorem 1
• proof
• Lemma 2
• Lemma 3
• Lemma 4
• Theorem 2
• proof
• proof