Size function

Size functions are shape descriptors, in a geometrical/topological sense.

to the natural numbers, counting certain connected components of a topological space.

They are used in pattern recognition and topology.

is equal to the number of connected components of the set

takes a value smaller than or equal to

[3] The concept of size function can be easily extended to the case of a measuring function

is endowed with the usual partial order .

[5] Size functions were introduced in [6] for the particular case of

equal to the topological space of all piecewise

closed manifold embedded in a Euclidean space.

equal to the topological space of all ordered

-tuples of points in a submanifold of a Euclidean space is considered.

An extension of the concept of size function to algebraic topology was made in [2] where the concept of size homotopy group was introduced.

Here measuring functions taking values in

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 [9] studied in persistent homology.

It is worth to point out that the size function is the rank of the

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.

[3][10][11][12][13][14][15][16][17] 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

is a compact locally connected Hausdorff space.

is a smooth closed manifold and

-function, the following useful property holds: A strong link between the concept of size function and the concept of natural pseudodistance

[1][19] 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.

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 [1] [20] .

[21] 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: 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.

An example of size function. (A) A size pair , where is the blue curve and is the height function. (B) The set is depicted in green. (C) The set of points at which the measuring function takes a value smaller than or equal to , that is, , is depicted in red. (D) Two connected components of the set contain at least one point in , that is, at least one point where the measuring function takes a value smaller than or equal to . (E) The value of the size function in the point is equal to .