Merge Tree Geodesics and Barycenters with Path Mappings

Abstract

Fig. 1: Two merge tree geodesics with their corresponding mappings. The top row shows the mappings of critical points (restricted to maxima) embedded into the domain. In the bottom row the mapping between the two corresponding merge trees and their interpolated geodesic is shown. The most persistent mapped features are highlighted and colored to show the correspondence between embedding and planar layout. The Wasserstein geodesic can be seen on the left, the path mapping geodesic on the right. Clearly, the middle tree on the right is a more meaningful interpolation of the two input trees than the middle tree on the left; and yields a more intuitive correspondence between the nodes. Note that the used layout differs between the two figures. However, the two trees in front are indeed identical in both figures, as are the trees in the back.