Merge Tree Geodesics and Barycenters with Path Mappings

TL;DR

The paper addresses the challenge of comparing scalar fields using topological descriptors by merging path-mapping distance with Wasserstein geodesics/barycenters on merge trees. It introduces deformation-based geodesics and barycenters built from path mappings, including strategies to handle non-pure edge mappings with imaginary nodes and to ensure structural validity. Empirical results across ensemble summarization, clustering, and time-series reduction show that path-mapping barycenters often yield higher-quality representatives and more accurate clustering than Wasserstein-based counterparts, albeit with higher computational costs. The work provides an open-source C++ implementation within the Topology ToolKit (TTK), enabling practical application and integration into existing visualization pipelines.

Abstract

Comparative visualization of scalar fields is often facilitated using similarity measures such as edit distances. In this paper, we describe a novel approach for similarity analysis of scalar fields that combines two recently introduced techniques: Wasserstein geodesics/barycenters as well as path mappings, a branch decomposition-independent edit distance. Effectively, we are able to leverage the reduced susceptibility of path mappings to small perturbations in the data when compared with the original Wasserstein distance. Our approach therefore exhibits superior performance and quality in typical tasks such as ensemble summarization, ensemble clustering, and temporal reduction of time series, while retaining practically feasible runtimes. Beyond studying theoretical properties of our approach and discussing implementation aspects, we describe a number of case studies that provide empirical insights into its utility for comparative visualization, and demonstrate the advantages of our method in both synthetic and real-world scenarios. We supply a C++ implementation that can be used to reproduce our results.

Figures (11)

• Figure 1: Example of barycenter assignment and update for member trees $T_1,T_2,T_3$ and initial candidate $T_1$. Optimal path mappings are illustrated through the edge colors. Edge lengths can be read from the grid.
• Figure 2: Example of the barycenter update: Three merge trees $T_1$, $T_2$ and $T_3$ and two path mappings (dashed lines) are shown. $T_2$ and $T_3$ contain an imaginary node, highlighted through the dotted strokes.
• Figure 3: The six member split trees of the starting vortex ensemble with the two barycenters on the right. Branches of low persistence are uncolored and drawn thinner. All member trees (a) and the path mapping barycenter (b) contain one edge of high persistence without a fork structure as well as a fork structure of slightly lower persistence. The long edge also forms the main branch of the branch decomposition in all but one of the member trees. In contrast to that, the Wasserstein barycenter (c) creates a fork structure within this main branch, thus having two high-persistence forks. The reason is that the fork structure is the main branch in one member which leads to bad mappings.
• Figure 4: Four member fields from the analytical example with their split trees, shown in (a). On the right, the two barycenters are shown. The barycenter in (b) was computed using the path mapping distance, the one in (c) using the Wasserstein distance.
• Figure 5: Two screenshots of the clustering output for the TOSCA ensemble in ParaView. The meshes are colored by their shape name (note that the human cluster consists of two different shapes) and arranged according to the assigned cluster. The screenshot in (b) shows a correct result, which assigns all human shapes into one cluster, as well as all centaurs and all seahorses. It was computed using the path mapping barycenters. The screenhot in (a) shows an incorrect clustering computed by the Wasserstein barycenters.

Theorems & Definitions (2)

• Definition 1
• Definition 2