Size functions are shape descriptors, in a geometrical/topological sense. They are functions from the half-plane
Contents
Formal definition
In size theory, the size function
History and applications
Size functions were introduced in for the particular case of
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
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
Main properties
Assume that
¤ every size function
¤ every size function
¤ for every
¤ for every
¤ for every
If we also assume that
¤ in order that
A strong link between the concept of size function and the concept of natural pseudodistance
¤ if
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:
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.