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A Structural Average of Labeled Merge Trees for Uncertainty Visualization.

Lin Yan, Yusu Wang, Elizabeth Munch

IEEE Transactions on Visualization and Computer Graphics

|August 13, 2019

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This study introduces a method to average labeled merge trees, encoding data uncertainty. The approach uses interleaving distances to find a central tree, aiding visualization of complex scalar field data.

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Area of Science:

Topological Data Analysis
Scientific Visualization
Uncertainty Quantification

Background:

Scalar fields are fundamental in modeling physical phenomena across science and engineering.
Graph-based descriptors like merge trees, contour trees, and Reeb graphs analyze topological changes in scalar fields.
Incorporating uncertainty into these topological descriptors is crucial for robust data analysis and visualization.

Purpose of the Study:

To develop a method for computing a structural average of labeled merge trees to represent and visualize data uncertainty.
To introduce a mathematically rigorous framework for understanding the structural relationships within a set of merge trees.
To create an interactive system for visualizing this structural average and associated uncertainty.

Main Methods:

Computation of a 1-center tree minimizing maximum distance to other trees using the interleaving distance metric.
Development of heuristic strategies for averaging merge trees with non-identical labels.
Introduction of a novel uncertainty measure, 'consistency', based on a metric-space view of the trees.
Creation of an interactive visualization system for computing and exploring structural averages.

Main Results:

A method for computing a mathematically rigorous structural average of merge trees is presented.
The framework effectively encodes uncertainty in scalar field data using merge tree ensembles.
An interactive system visualizes the structural average, highlighting similarities and uncertainty.

Conclusions:

The proposed framework provides a novel approach to uncertainty visualization in scalar fields using topological descriptors.
The use of interleaving distances and consistency offers a robust way to analyze structural averages of merge trees.
This work advances topology-based visualization by integrating uncertainty quantification.