Categorizing Merge Tree Edit Distances by Stability using Minimal Vertex Perturbation

TL;DR

This paper proposes a new stability measure for merge tree edit distances, enabling better classification and understanding of scalar field comparison metrics in scientific visualization.

Contribution

It introduces a novel stability measure for merge tree edit distances, addressing limitations of previous measures and aligning with practical implementation efficiency.

Findings

New stability measure effectively classifies merge tree distances.

The measure captures fine-grained hierarchy of scalar field comparison metrics.

Results highlight open questions in theoretical analysis of practical distances.

Abstract

This paper introduces a novel stability measure for edit distances between merge trees of piecewise linear scalar fields. We apply the new measure to various metrics introduced recently in the field of scalar field comparison in scientific visualization. While previous stability measures are unable to capture the fine-grained hierarchy of the considered distances, we obtain a classification of stability that fits the efficiency of current implementations and quality of practical results. Our results induce several open questions regarding the lacking theoretical analysis of such practical distances.