ParkView: Visualizing Monotone Interleavings

2025-01-18

AI Summary

Visualizing similarity between merge trees in topological data analysis remains challenging due to the difficulty of intuitively representing interleaving structures under leaf-order constraints. Method: We propose a structured visual encoding framework grounded in monotone interleavings. First, we establish a path-branch optimal decomposition theory for monotone interleavings, revealing their intrinsic sparsity. Building on this, we introduce the *hedge encoding* paradigm—a topology-aware visual scheme that unambiguously maps active paths and monotone mappings using at most six colors. Our method integrates linear-time path-branch decomposition, graph-theoretic modeling, and topologically constrained visual encoding. Results: The approach enables efficient, interpretable visualization of merge trees of arbitrary size. Extensive evaluation on real-world datasets demonstrates significant advantages in compactness, readability, and computational feasibility—achieving scalable, theoretically grounded visual analysis of merge tree similarity.

Technology Category

• Data Mining & Knowledge Management: Data Visualization & Summarization
• Knowledge Representation and Reasoning: Nonmonotonic Reasoning
• Computer Vision: Segmentation

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Web Mining and Content Analysis: Web data visualization
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics

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.

Problem

• Visualization
• Monotonic Interleaving
• Tree Similarity

Innovation

• ParkView
• Visual Simplification
• Monotone Interleaving Comparison