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

Florian Wetzels, Markus Anders, Christoph Garth

TL;DR

This work tackles instability in comparing scalar fields via merge trees by introducing the unconstrained deformation-based edit distance, which handles both vertical instabilities and saddle swaps (horizontal instabilities). It provides an $NP$-complete decision-theoretic hardness result, an integer programming formulation with optimization strategies, and empirical evidence from synthetic tests and the TOSCA dataset showing enhanced stability and clearer clustering relative to prior methods. The contribution includes a rigorous complexity result, a practical computation framework, and extensive discussion on stability and future improvements. Overall, the approach advances robust, topology-based scalar-field comparison by enabling saddle-swaps-aware edit distances for merge trees.

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.

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

TL;DR

This work tackles instability in comparing scalar fields via merge trees by introducing the unconstrained deformation-based edit distance, which handles both vertical instabilities and saddle swaps (horizontal instabilities). It provides an -complete decision-theoretic hardness result, an integer programming formulation with optimization strategies, and empirical evidence from synthetic tests and the TOSCA dataset showing enhanced stability and clearer clustering relative to prior methods. The contribution includes a rigorous complexity result, a practical computation framework, and extensive discussion on stability and future improvements. Overall, the approach advances robust, topology-based scalar-field comparison by enabling saddle-swaps-aware edit distances for merge trees.

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.
Paper Structure (7 sections, 1 theorem, 8 equations, 12 figures)

This paper contains 7 sections, 1 theorem, 8 equations, 12 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Computation
  4. NP-completeness
  5. Stability
  6. Experiments
  7. Conclusion and Future Work

Key Result

Theorem 1

Deciding whether $\delta_{\text{E}}(T_1,T_2)$ of two abstract merge trees $T_1,T_2$ is below a given threshold $c$ is -complete.

Figures (12)

  • Figure 1: Intuition behind the deformation-based edit operations: if a degree-1 node remains after a deletion, we obtain an invalid merge tree. To fix this problem, we also have to prune the remaining node.
  • Figure 2: The element gadgets $G$ and $G_i$.
  • Figure 3: The dummy set gadget $H$ and set gadget $H_j$.
  • Figure 4: The two trees used in the reduction.
  • Figure 5: The two types of instabilities: In (a) the order of saddle points changes such that feature $C$ emerges from $B$ instead of $A$. Thus, we also call this a saddle swap. In (b) the two features $x$ and $y$ swap their position in the persistence-based ordering. Thus, mappings adhering to this hierarchy cannot map $y$ to $y$ and, more importantly, not map the subtrees on the path to $y$ to each other.
  • ...and 7 more figures

Theorems & Definitions (3)

  • Definition : Abstract Merge Tree
  • Theorem 1
  • proof