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'''Size functions''' are shape descriptors, in a geometrical/topological sense. They are functions from the half-plane <math>x<y</math> to the natural numbers, counting certain [[Connected component (topology)|connected
==Formal definition==
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and the size homotopy group is analogous to the one existing between [[homology group]]s and [[homotopy group]]s.
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,
''On the use of size functions for shape analysis'',
Biological Cybernetics, 70:99–107, 1993.</ref><ref>Patrizio Frosini and Claudia Landi,
''Size functions and morphological transformations'',
Acta Applicandae Mathematicae, 49(1):85–104, 1997.</ref><ref>Alessandro Verri and Claudio Uras,
''Metric-topological approach to shape
representation and recognition'',
Image Vision Comput., 14:189–207, 1996.</ref><ref>Alessandro Verri and Claudio Uras,
''Computing size functions from edge maps'',
Internat. J. Comput. Vision, 23(2):169–183, 1997.</ref><ref>Françoise Dibos, Patrizio Frosini and Denis Pasquignon,
''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="CeFeGi06">Andrea Cerri, Massimo Ferri, Daniela Giorgi, ''Retrieval of trademark images by means of size functions Graphical Models'' 68:451–471, 2006.</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>▼
▲.<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>.
==Main properties==
Assume that <math>M</math> is a compact [[locally connected]] [[Hausdorff space]]. The following statements hold:
* every size function <math>\ell_{(M,\varphi)}(x,y)</math> is a [[non-decreasing function]] in the variable <math>x</math> and a [[Nonincreasing function|non-increasing function]] in the variable <math>y</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]]
<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>
* 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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* ''cornerpoints'' are defined as those points <math>p=(x,y)</math>, with <math>x<y</math>, such that the number
::<math>\mu (p){\stackrel{{\rm def}}{=}}\min _{\alpha >0 ,\beta>0} \ell _{({M},\varphi )}(x+\alpha ,y-
\beta)-\ell _{({ M},\varphi )} (x+\alpha ,y+\beta )-
\ell_{({ M},\varphi )} (x-\alpha ,y-\beta )+\ell _{({ M}
,\varphi )} (x-\alpha ,y+\beta )</math>
:is positive. The number <math>\mu (p)</math> is said to be the ''multiplicity'' of <math>p</math>.▼
▲The number <math>\mu (p)</math> is said to be the ''multiplicity'' of <math>p</math>.
* ''cornerlines'' and are defined as those lines <math>r:x=k</math> such that
:: <math>\mu (r){\stackrel{\rm def}{=}}\min _{\alpha >0 ,k+\alpha <y}\ell _{({ M},\varphi
)}(k+\alpha ,y)-
\ell _{({ M},\varphi )}(k-\alpha ,y)>0.</math>
: The number <math>\mu (r)</math> is sad to be the '' multiplicity'' of <math>r</math>.
* ''Representation Theorem'': For every <math>{\bar x}<{\bar y}</math>, it holds
This representation contains the
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