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.

翻译：科学可视化中标量场的比较分析通常涉及拓扑抽象的距离函数。本文聚焦于合并树抽象（表示子/超水平集嵌套结构），并提出采用无约束形变编辑距离。现有合并树方法常存在不稳定性问题：数据中的微小扰动可能导致抽象距离显著增大。尽管部分方法能处理所谓的垂直不稳定性，但无约束形变编辑距离可同时解决垂直与水平不稳定性（即鞍点交换）。我们证明其计算复杂度为NP完全问题，并给出整数线性规划求解方案。在TOSCA形状匹配数据集上的实验结果表明该距离具有稳定性，从而展示了通过合并树处理鞍点交换以比较标量场的潜力。