ParkView: Visualizing Monotone Interleavings

TL;DR

ParkView is a scalable visualization method for monotone interleavings of merge trees, using optimal decompositions and color coding to clearly depict complex topological relationships in scalar field analysis.

Contribution

The paper introduces ParkView, a novel, efficient encoding for visualizing monotone interleavings with structural guarantees and linear-time computation.

Findings

Supports clear visualization of complex interleavings

Uses optimal path-branch decomposition for scalability

Achieves effective coloring scheme for large trees

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.