Lifted Wasserstein Matcher for Fast and Robust Topology Tracking

TL;DR

This paper introduces a fast, robust method for tracking topological features in time-varying data using an improved Wasserstein metric and optimized assignment algorithms, enabling efficient and accurate topology tracking.

Contribution

It adapts the Kuhn-Munkres assignment algorithm for persistence diagram matching and proposes a geometrical extension of the Wasserstein metric to enhance stability and efficiency.

Findings

Our method is an order of magnitude faster than the Munkres algorithm.

It provides competitive runtimes with modern approximation methods while maintaining exact results.

The approach is robust to noise and temporal downsampling.

Abstract

This paper presents a robust and efficient method for tracking topological features in time-varying scalar data. Structures are tracked based on the optimal matching between persistence diagrams with respect to the Wasserstein metric. This fundamentally relies on solving the assignment problem, a special case of optimal transport, for all consecutive timesteps. Our approach relies on two main contributions. First, we revisit the seminal assignment algorithm by Kuhn and Munkres which we specifically adapt to the problem of matching persistence diagrams in an efficient way. Second, we propose an extension of the Wasserstein metric that significantly improves the geometrical stability of the matching of domain-embedded persistence pairs. We show that this geometrical lifting has the additional positive side-effect of improving the assignment matrix sparsity and therefore computing time. The global framework implements a coarse-grained parallelism by computing persistence diagrams and finding optimal matchings in parallel for every couple of consecutive timesteps. Critical trajectories are constructed by associating successively matched persistence pairs over time. Merging and splitting events are detected with a geometrical threshold in a post-processing stage. Extensive experiments on real-life datasets show that our matching approach is an order of magnitude faster than the seminal Munkres algorithm. Moreover, compared to a modern approximation method, our method provides competitive runtimes while yielding exact results. We demonstrate the utility of our global framework by extracting critical point trajectories from various simulated time-varying datasets and compare it to the existing methods based on associated overlaps of volumes. Robustness to noise and temporal resolution downsampling is empirically demonstrated.