We present a method to find repeating topological structures in scalar data sets. More precisely, we compare all subtrees of two merge trees against each other -in an efficient manner exploiting redundancy. This provides pair-wise distances between the topological structures defined by sub/superlevel sets, which can be exploited in several applications such as finding similar structures in the same data set, assessing periodic behavior in time-dependent data, and comparing the topology of two different data sets. To do so, we introduce a novel data structure called the extended branch decomposition graph, which is composed of the branch decompositions of all subtrees of the merge tree. Based on dynamic programming, we provide two highly efficient algorithms for computing and comparing extended branch decomposition graphs. Several applications attest to the utility of our method and its robustness against noise.

We present an algorithm for tracking regions in time‐dependent scalar fields that uses global knowledge from all time steps for determining the tracks. The regions are defined using merge trees, thereby representing a hierarchical segmentation of the data in each time step. The similarity of regions of two consecutive time steps is measured using their volumetric overlap and a histogram difference. The main ingredient of our method is a directed acyclic graph that records all relevant similarity information as follows: the regions of all time steps are the nodes of the graph, the edges represent possible short feature tracks between consecutive time steps, and the edge weights are given by the similarity of the connected regions. We compute a feature track as the global solution of a shortest path problem in the graph. We use these results to steer the – to the best of our knowledge – first algorithm for spatio‐temporal feature similarity estimation. Our algorithm works for 2D and 3D time‐dependent scalar fields. We compare our results to previous work, showcase its robustness to noise, and exemplify its utility using several real‐world data sets.

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