Yan, LinMasood, Talha BinSridharamurthy, RaghavendraRasheed, FarhanNatarajan, VijayHotz, IngridWang, BeiSmit, Noeska and Vrotsou, Katerina and Wang, Bei2021-06-122021-06-1220211467-8659https://doi.org/10.1111/cgf.14331https://diglib.eg.org:443/handle/10.1111/cgf14331In 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.10.1111/cgf.14331599-633