Size function: Difference between revisions

Revision as of 07:05, 13 January 2023

Revision as of 00:06, 4 April 2023

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==Formal definition==

In [[size theory]], the '''size function''' <math>\ell_{(M,\varphi)}:\Delta^+=\{(x,y)\in \mathbb{R}^2:x<y\}\to \mathbb{N}</math> associated with the [[size pair]] <math>(M,\varphi:M\to \mathbb{R})</math> is defined in the following way. For every <math>(x,y)\in \Delta^+</math>, <math>\ell_{(M,\varphi)}(x,y)</math> is equal to the number of connected components of the set

<math>\{p\in M:\varphi(p)\le y\}</math> that contain at least one point at which the [[measuring function]] (a [[continuous function]] from a [[topological space]] <math>M</math> to <math>\mathbb{R}^k</math>

<ref name="FroLa99">Patrizio Frosini and Claudia Landi, ''Size Theory as a Topological Tool for Computer Vision'', Pattern Recognition And Image Analysis, 9(4):596–603, 1999.</ref>

<ref name="FroMu99">Patrizio Frosini and Michele Mulazzani, ''Size homotopy groups for computation of natural size distances'', [[Bulletin of the Belgian Mathematical Society]], 6:455–464 1999.</ref>) <math>\varphi</math> takes a value smaller than or equal to <math>x</math>

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An extension to [[homology theory]] (the [[size functor]]) was introduced in

.<ref name="CaFePo01">Francesca Cagliari, Massimo Ferri and Paola Pozzi, ''Size functions from a categorical viewpoint'', Acta Applicandae Mathematicae, 67(3):225–235, 2001.</ref>

The concepts of [[size homotopy group]] and [[size functor]] are strictly related to the concept of [[persistent homology group]]

<ref name="EdLeZo02">Herbert Edelsbrunner, David Letscher and Afra Zomorodian, ''Topological Persistence and Simplification'', [[Discrete and Computational Geometry]], 28(4):511–533, 2002.</ref>

studied in [[persistent homology]]. It is worth to point out that the size function is the rank of the <math>0</math>-th persistent homology group, while the relation between the persistent homology group