  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.
