# A Structural Average of Labeled Merge Trees for Uncertainty Visualization

## TL;DR

This paper introduces a method to compute a structural average of labeled merge trees to encode and visualize uncertainty in scalar field data, using interleaving distance and consistency measures.

## Contribution

It presents a novel approach to aggregate merge trees under uncertainty, employing interleaving distance and heuristic algorithms for structural averaging.

## Key findings

- Developed a 1-center tree that minimizes maximum distance to input trees.
- Created an interactive visualization system for merge tree averaging.
- Demonstrated the approach on ensembles of scalar fields.

## Abstract

Physical phenomena in science and engineering are frequently modeled using scalar fields. In scalar field topology, graph-based topological descriptors such as merge trees, contour trees, and Reeb graphs are commonly used to characterize topological changes in the (sub)level sets of scalar fields. One of the biggest challenges and opportunities to advance topology-based visualization is to understand and incorporate uncertainty into such topological descriptors to effectively reason about their underlying data. In this paper, we study a structural average of a set of labeled merge trees and use it to encode uncertainty in data. Specifically, we compute a 1-center tree that minimizes its maximum distance to any other tree in the set under a well-defined metric called the interleaving distance. We provide heuristic strategies that compute structural averages of merge trees whose labels do not fully agree. We further provide an interactive visualization system that resembles a numerical calculator that takes as input a set of merge trees and outputs a tree as their structural average. We also highlight structural similarities between the input and the average and incorporate uncertainty information for visual exploration. We develop a novel measure of uncertainty, referred to as consistency, via a metric-space view of the input trees. Finally, we demonstrate an application of our framework through merge trees that arise from ensembles of scalar fields. Our work is the first to employ interleaving distances and consistency to study a global, mathematically rigorous, structural average of merge trees in the context of uncertainty visualization.