Size Function
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Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. Size functions are shape descriptors, in a geometrical/topological sense. They are functions from the half-plane xy to the natural numbers, counting certain connected components of a topological space. They are used in pattern recognition and topology. 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 mathbb{R}^k are allowed. An extension to...
Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. Size functions are shape descriptors, in a geometrical/topological sense. They are functions from the half-plane xy to the natural numbers, counting certain connected components of a topological space. They are used in pattern recognition and topology. 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 mathbb{R}^k are allowed. An extension to homology theory (the size functor) was introduced in [8]. 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 0-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.
