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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="FroMu99">Patrizio Frosini, 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> A survey about size functions (and [[size theory]]) can be found in
''Describing shapes by geometrical-topological properties of real functions''
ACM Computing Surveys, vol. 40 (2008), n. 4, 12:1–12:87.</ref>.
[[File:SFesWiki.PNG|1065px|thumb|left|''An example of size function. (A) A size pair <math>(M,\varphi:M\to\mathbb{R})</math>, where <math>M</math> is the blue curve and <math>\varphi:M\to \mathbb{R}</math> is the height function. (B) The set <math>\{p\in M:\varphi(p)\le b\}</math> is depicted in green. (C) The set of points at which the [[measuring function]] <math>\varphi</math> takes a value smaller than or equal to <math>a</math>, that is, <math>\{p\in M:\varphi(p)\le a\}</math>, is depicted in red. (D) Two connected component of the set <math>\{p\in M:\varphi(p)\le b\}</math> contain at least one point in <math>\{p\in M:\varphi(p)\le a\}</math>, that is, at least one point where the [[measuring function]] <math>\varphi</math> takes a value smaller than or equal to <math>a</math>. (E) The value of the size function <math>\ell_{(M,\varphi)}</math> in the point <math>(a,b)</math> is equal to <math>2</math>.'']]
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where the concept of [[size homotopy group]] was introduced. Here [[measuring function]]s taking values in <math>\mathbb{R}^k</math> 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]]
<ref name="EdLeZo02">Herbert Edelsbrunner, David Letscher and Afra Zomorodian, ''Topological Persistence and Simplification'', [[Discrete and Computational Geometry]], 28(4):511–533, 2002.</ref>
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Journal of Mathematical Imaging and Vision, 21(2):107–118, 2004.</ref>
<ref name="CeFeGi06">Andrea Cerri, Massimo Ferri, Daniela Giorgi, ''Retrieval of trademark images by means of size functions Graphical Models'' 68:451–471, 2006.</ref>
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 <math>\Delta^+\ </math>.
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If we also assume that <math>M\ </math> is a smooth [[closed manifold]] and <math>\varphi</math> is a <math>C^1\ </math>-function, the following useful property holds:
¤ in order that <math>(x,y)\ </math> is a discontinuity point for <math>\ell_{(M,\varphi)}</math> it is necessary that either <math>x\ </math> or <math>y\ </math> or both are critical values for <math>\varphi</math>
<ref name="Fro96">Patrizio Frosini, ''Connections between size functions and critical points'', Mathematical Methods In The Applied Sciences, 19:555–569, 1996.</ref>.
A strong link between the concept of size function and the concept of [[natural pseudodistance]]
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<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="LaFro97">Claudia Landi and Patrizio Frosini, ''New pseudodistances for the size function space'', Proc. SPIE Vol. 3168, p. 52-60, Vision Geometry VI, Robert A. Melter, Angela Y. Wu, Longin J. Latecki (eds.), 1997.</ref>
The points (called ''cornerpoints'') and lines (called ''cornerlines'') of such formal series encode the information about
discontinuities of the corresponding size functions, while
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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==
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