Describing shapes by geometrical- topological properties of real functions

In the last decade, an increasing number of methods that are rooted in Morse theory andmake use of properties of real-valued functions for describing shapes have been proposed in theliterature. The methods proposed range from approaches which use the configuration of contoursfor encoding topographic surfaces to more recent work on size theory and persistent homology.All these have been developed over the years with a specific target domain and it is not trivial tosystematize this work and understand the links, similarities and differences among the differentmethods. Moreover, different terms have been used to denote the same mathematical constructs,which often overwhelms 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, focusingon methods that make use of theoretical frameworks that are developed for classes of realfunctions rather than for a single function, even if they are applied in a restricted manner. Theterm geometrical-topological used in the title is meant to underline that both levels of informationcontent are relevant for the applications of shape descriptions: geometrical, or metrical, propertiesand attributes are crucial for characterizing specific instances of features, while topologicalproperties are necessary to abstract and classify shapes according to invariant aspects of their geometry.The approaches surveyed will be discussed in detail, with respect to theory, computationand application. Several properties of the shape descriptors will be analyzed and compared. Webelieve this is a crucial step to exploit fully the potential of such approaches in many applications,as well as to identify important areas of future research.