Describing shapes by geometrical-topological properties of real functions

An increasing number of methods that are rooted in Morse theory and make use of properties of real-valued functions for describing shapes have been proposed in the literature. The methods range from approaches which use the configuration of contours for encoding topographic surfaces to more recent work on size theory and persistent homology. It is not trivial to systematize this work and understand the links, similarities, and differences among the different methods. Moreover, different terms have been used to denote the same constructs, which often overwhelm the understanding of the underlying common framework. The aim of this survey is to provide a clear vision of what has been developed so far. The approaches surveyed are analysed, discussed and compared in detail, with respect to theory, computation, and application. We believe this is a crucial step to fully exploit the potential of such approaches, as well as to identify important areas of future research.