Fig. 1. The Persistence diagrams of three members (a-c) of the Isabel ensemble (wind velocity) concisely and visually encode the number, data range and salience of the features of interest found in the data (eyewall and region of high speed wind, blue and red in (a)). In these diagrams, features with a persistence smaller than 10% of the function range or on the boundary are shown in transparent white. The pointwise mean for these three members (d) exhibits three salient interior features (due to distinct eyewall locations, blue, green and red), although the diagrams of the input members only report two salient interior features at most, located at drastically different data ranges (the red feature is further down the diagonal in (a) and (b)). The Wasserstein barycenter of these three diagrams (e) provides a more representative view of the features found in this ensemble, as it reports a feature number, range and salience that better matches the input diagrams (a-c). Our work introduces a progressive approximation algorithm for such barycenters, with fast practical convergence. Our framework supports computation time constraints (e) which enables the approximation of Wasserstein barycenters within interactive times. We present an application to the clustering of ensemble members based on their persistence diagrams ((f), lifting: α = 0.2), which enables the visual exploration of the main trends of features of interest found in the ensemble.Abstract-This paper presents an efficient algorithm for the progressive approximation of Wasserstein barycenters of persistence diagrams, with applications to the visual analysis of ensemble data. Given a set of scalar fields, our approach enables the computation of a persistence diagram which is representative of the set, and which visually conveys the number, data ranges and saliences of the main features of interest found in the set. Such representative diagrams are obtained by computing explicitly the discrete Wasserstein barycenter of the set of persistence diagrams, a notoriously computationally intensive task. In particular, we revisit efficient algorithms for Wasserstein distance approximation [12,51] to extend previous work on barycenter estimation [94]. We present a new fast algorithm, which progressively approximates the barycenter by iteratively increasing the computation accuracy as well as the number of persistent features in the output diagram. Such a progressivity drastically improves convergence in practice and allows to design an interruptible algorithm, capable of respecting computation time constraints. This enables the approximation of Wasserstein barycenters within interactive times. We present an application to ensemble clustering where we revisit the k-means algorithm to exploit our barycenters and compute, within execution time constraints, meaningful clusters of ensemble data along with their barycenter diagram. Extensive experiments on synthetic and real-life data sets report that our algorithm converges to barycenters that are qualitatively meaningfu...

This software paper gives an overview of the features supported by the Topology ToolKit (TTK), which is an open-source library for topological data analysis (TDA). TTK implements, in a generic and efficient way, a substantial collection of reference algorithms in TDA. Since its initial public release in 2017, both its user and developer bases have grown, resulting in a significant increase in the number of supported features. In contrast to the original paper introducing TTK [40] (which detailed the core algorithms and data structures of TTK), the purpose of this software paper is to describe the list of features currently supported by TTK, ranging from image segmentation tools to advanced topological analysis of high-dimensional data, with concrete usage examples available on the TTK website [42].