Size function

Size functions are shape descriptors, in a geometrical/topological sense. They are functions from the half-plane to the natural numbers, counting certain connected components of a topological space. They are used in pattern recognition and topology. In size theory, the size function associated with the size pair is defined in the following way. For every , is equal to the number of connected components of the set that contain at least one point at which the measuring function (a continuous function from a topological space to ) takes a value smaller than or equal to The concept of size function can be easily extended to the case of a measuring function , where is endowed with the usual partial order A survey about size functions (and size theory) can be found in. Size functions were introduced in for the particular case of equal to the topological space of all piecewise closed paths in a closed manifold embedded in a Euclidean space. Here the topology on is induced by the norm, while the measuring function takes each path to its length. In the case of equal to the topological space of all ordered -tuples of points in a submanifold of a Euclidean space is considered. Here the topology on is induced by the metric . An extension of the concept of size function to algebraic topology was made in where the concept of size homotopy group was introduced. Here measuring functions taking values in are allowed. An extension to homology theory (the size functor) was introduced in The concepts of size homotopy group and size functor are strictly related to the concept of persistent homology group studied in persistent homology. It is worth to point out that the size function is the rank of the -th persistent homology group, while the relation between the persistent homology group and the size homotopy group is analogous to the one existing between homology groups and homotopy groups. Size functions have been initially introduced as a mathematical tool for shape comparison in computer vision and pattern recognition, and have constituted the seed of size theory.