Size function

Size functions are shape descriptors, in a geometrical/topological sense. They are functions from the half-plane formula_1 to the natural numbers, counting certain connected components of a topological space. They are used in pattern recognition and topology.
Formal definition.
In size theory, the size function formula_2 associated with the size pair formula_3 is defined in the following way. For every formula_4, formula_5 is equal to the number of connected components of the set 
formula_6 that contain at least one point at which the measuring function (a continuous function from a topological space formula_7 to formula_8
) formula_9 takes a value smaller than or equal to formula_10
The concept of size function can be easily extended to the case of a measuring function formula_11, where formula_8 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
for the particular case of formula_7 equal to the topological space of all piecewise formula_14 closed paths in a formula_15 closed manifold embedded in a Euclidean space. Here the topology on formula_7 is induced by the
formula_17-norm, while the measuring function formula_9 takes each path formula_19 to its length.
In
the case of formula_7 equal to the topological space of all ordered formula_21-tuples of points in a submanifold of a Euclidean space is considered.
Here the topology on formula_7 is induced by the metric formula_23.
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 formula_8 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 formula_25-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 formula_26.
Main properties.
Assume that formula_7 is a compact locally connected Hausdorff space. The following statements hold:
If we also assume that formula_7 is a smooth closed manifold and formula_9 is a formula_14-function, the following useful property holds:
A strong link between the concept of size function and the concept of natural pseudodistance
formula_51 between the size pairs formula_52 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:
formula_57 
is positive.
The number formula_58 is said to be the "multiplicity" of formula_59.
formula_61
The number formula_62 is sad to be the " multiplicity" of formula_63.
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.