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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, 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, 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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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 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]]
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
and the size homotopy group is analogous to the one existing between [[homology group]]s and [[homotopy group]]s.
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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]]
<ref> Claudio Uras and Alessandro Verri, ''[http://www.icsi.berkeley.edu/pubs/techreports/tr-92-057.pdf Describing and recognising shape through size functions]'' ICSI Technical Report TR-92-057, Berkeley, 1992.</ref>
''On the use of size functions for shape analysis'',
Biological Cybernetics, 70:99–107, 1993.</ref>
''Size functions and morphological transformations'',
Acta Applicandae Mathematicae, 49(1):85–104, 1997.</ref>
''Metric-topological approach to shape
representation and recognition'',
Image Vision Comput., 14:189–207, 1996.</ref>
''Computing size functions from edge maps'',
Internat. J. Comput. Vision, 23(2):169–183, 1997.</ref>
''The use of size functions for comparison of shapes through differential invariants'',
Journal of Mathematical Imaging and Vision, 21(2):107–118, 2004.</ref>
<ref name="BiGiSp08">Silvia Biasotti, Daniela Giorgi, Michela Spagnuolo, Bianca Falcidieno, ''Size functions for comparing 3D models'' Pattern Recognition 41:2855–2873, 2008.</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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A strong link between the concept of size function and the concept of [[natural pseudodistance]]
<math>d((M,\varphi),(N,\psi))</math> between the size pairs <math>(M,\varphi),\ (N,\psi)</math> exists
¤ if <math>\ell_{(N,\psi)}(\bar x,\bar y)>\ell_{(M,\varphi)}(\tilde x,\tilde y)</math> then <math>d((M,\varphi),(N,\psi))\ge \min\{\tilde x-\bar x,\bar y-\tilde y\}</math>.
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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
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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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