  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.
