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<math>C^0\ </math>-norm, while the [[measuring function]] <math>\varphi\ </math> takes each path <math>\gamma\in M</math> to its length.
In
<ref name="Fro91">Patrizio Frosini, ''Measuring shapes by size functions'', Proc.
the case of <math>M\ </math> equal to the topological space of all ordered <math>k\ </math>-tuples of points in a submanifold of a Euclidean space is considered.
Here the topology on <math>M\ </math> is induced by the metric <math>d((P_1,\ldots,P_k),(Q_1\ldots,Q_k))=\max_{1\le i\le k}\|P_i-Q_i\|</math>.
An extension of the concept of size function to [[algebraic topology]] was made in
<ref name="FroMu99"/>
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
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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 name="dAFrLa06"/>
<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>
<ref>Alessandro Verri, Claudio Uras, Patrizio Frosini and Massimo Ferri,
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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
<ref name="FroLa99"/>
<ref name="DoFro04">Pietro Donatini and Patrizio Frosini, ''Lower bounds for natural pseudodistances via size functions'', Archives of Inequalities and Applications, 2(1):1–12, 2004.</ref>
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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
<ref name="FroLa99"/>
<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>
.<ref name="FroLa01">Patrizio Frosini and Claudia Landi, ''Size functions and formal series'', Appl. Algebra Engrg. Comm. Comput., 12:327–349, 2001.</ref>
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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]].<ref name="dAFrLa06"
==References==
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