Taming Horizontal Instability in Merge Trees: On the Computation of a Comprehensive Deformation-based Edit Distance

TL;DR

This paper introduces an unconstrained deformation-based edit distance for merge trees that effectively handles both vertical and horizontal instabilities, including saddle swaps, improving the robustness of scalar field comparisons.

Contribution

It presents a novel deformation-based edit distance that addresses horizontal instability in merge trees and provides an ILP formulation for its computation.

Findings

The proposed distance is stable against small data perturbations.

It effectively handles saddle swaps in merge trees.

Experimental results demonstrate improved robustness in shape matching.

Abstract

Comparative analysis of scalar fields in scientific visualization often involves distance functions on topological abstractions. This paper focuses on the merge tree abstraction (representing the nesting of sub- or superlevel sets) and proposes the application of the unconstrained deformation-based edit distance. Previous approaches on merge trees often suffer from instability: small perturbations in the data can lead to large distances of the abstractions. While some existing methods can handle so-called vertical instability, the unconstrained deformation-based edit distance addresses both vertical and horizontal instabilities, also called saddle swaps. We establish the computational complexity as NP-complete, and provide an integer linear program formulation for computation. Experimental results on the TOSCA shape matching ensemble provide evidence for the stability of the proposed distance. We thereby showcase the potential of handling saddle swaps for comparison of scalar fields through merge trees.