Size functions to the natural numbers, counting certain connected componentss of a topological space. They are used in pattern recognition and topology.

Formal definition

In size theory, the size function associated with the size pair that contain at least one point at which the measuring function (a continuous function from a topological space. is endowed with the usual partial order. A survey about size functions (and size theory) can be found in.

History and applications

Size functions were introduced in closed manifold embedded in a Euclidean space-norm, while the measuring function to its length. In-tuples of points in a submanifold.

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 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-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.

Main properties

is a compact locally connected Hausdorff space. The following statements hold:

• is a non-decreasing function and a non-increasing function.

• is locally right-constant in both its variables.

• is finite.

• .

• .

is a smooth closed manifold-function, the following useful property holds:

• .

A strong link between the concept of size function and the concept of natural pseudodistance exists.

• .

The previous result gives an easy way to get lower bounds for the natural pseudodistance and is one of the main motivation to introduce the concept of size function.

Representation by formal series

An algebraic representation of size functions in terms of collections of points and lines in the real plane with multiplicities, i.e. as particular formal series, was furnished in. The points (called cornerpoints) and lines (called cornerlines) of such formal series encode the information about discontinuities of the corresponding size functions, while their multiplicities contain the information about the values taken by the size function.

Formally:

• cornerpoints, such that the number

• is said to be the multiplicity.

• cornerlines such that

• is sad to be the multiplicity.

• Representation Theorem, it holds

  1. .

This representation contains the same amount of information about the shape under study as the original size function does, but is much more concise.

This algebraic approach to size functions leads to the definition of new similarity measures between shapes, by translating the problem of comparing size functions into the problem of comparing formal series. The most studied among these metrics between size function is the matching distance.

References

See also

• Size theory

• Natural pseudodistance

• Size functor

• Size homotopy group

• Size pair

• Matching distance

• Topological data analysis