Scalar Field Comparison with Topological Descriptors: Properties and Applications for Scientific Visualization

In topological data analysis and visualization, topological descriptors such as persistence diagrams, merge trees, contour trees, Reeb graphs, and Morse–Smale complexes play an essential role in capturing the shape of scalar field data. We present a state-of-the-art report on scalar field comparison using topological descriptors. We provide a taxonomy of existing approaches based on visualization tasks associated with three categories of data: single fields, time-varying fields, and ensembles. These tasks include symmetry detection, periodicity detection, key event/feature detection, feature tracking, clustering, and structure statistics. Our main contributions include the formulation of a set of desirable mathematical and computational properties of comparative measures, and the classification of visualization tasks and applications that are enabled by these measures.Funding: United States Department of Energy (DOE)United States Department of Energy (DOE) [DE-SC0021015]; National Science Foundation (NSF)National Science Foundation (NSF) [IIS-1910733]; Indo-Swedish joint network project [DST/INT/SWD/VR/P-02/2019]; Swedish Research Council (VR)Swedish Research Council [2018-07085]; VR grant [2019-05487]; MHRD, Swarnajayanti Fellowship from the Department of Science and Technology, India [DST/SJF/ETA-02/2015-16]; Mindtree Chair research grant