Merge Tree Geodesics and Barycenters with Path Mappings

TL;DR

This paper introduces a new method combining Wasserstein geodesics, barycenters, and path mappings for improved scalar field similarity analysis, enhancing performance in visualization tasks with practical runtimes.

Contribution

It presents a novel approach that integrates path mappings with Wasserstein techniques, reducing sensitivity to data perturbations and improving comparative visualization.

Findings

Superior performance in ensemble summarization and clustering

Effective temporal reduction of time series data

Demonstrated advantages on synthetic and real-world data

Abstract

Comparative visualization of scalar fields is often facilitated using similarity measures such as edit distances. In this paper, we describe a novel approach for similarity analysis of scalar fields that combines two recently introduced techniques: Wasserstein geodesics/barycenters as well as path mappings, a branch decomposition-independent edit distance. Effectively, we are able to leverage the reduced susceptibility of path mappings to small perturbations in the data when compared with the original Wasserstein distance. Our approach therefore exhibits superior performance and quality in typical tasks such as ensemble summarization, ensemble clustering, and temporal reduction of time series, while retaining practically feasible runtimes. Beyond studying theoretical properties of our approach and discussing implementation aspects, we describe a number of case studies that provide empirical insights into its utility for comparative visualization, and demonstrate the advantages of our method in both synthetic and real-world scenarios. We supply a C++ implementation that can be used to reproduce our results.