Size Function - History and Applications

History and Applications

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, The main point is that size functions are invariant for every transformation preserving the measuring function. Hence, they can be adapted to many different applications, by simply changing the measuring function in order to get the wanted invariance. Moreover, size functions show properties of relative resistance to noise, depending on the fact that they distribute the information all over the half-plane .