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{{Short description|Shape descriptions in a geometrical/topological sense}}
{{Use dmy dates|date=July 2022}}
'''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 component]]s of a [[topological space]]. They are used in [[pattern recognition]] and [[topology]].
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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
<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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The concept of size function can be easily extended to the case of a measuring function <math>\varphi:M\to \mathbb{R}^k</math>, where <math>\mathbb{R}^k</math> is endowed with the usual partial order
.<ref name="BiCeFr08">Silvia Biasotti, Andrea Cerri, Patrizio Frosini, Claudia Landi, ''Multidimensional size functions for shape comparison'', Journal of Mathematical Imaging and Vision 32:161–179, 2008.</ref>
A survey about size functions (and [[size theory]]) can be found in.<ref name="BiDeFa08">Silvia Biasotti, [[Leila De Floriani]], [[Bianca Falcidieno]], Patrizio Frosini, Daniela Giorgi, Claudia Landi, Laura Papaleo, Michela Spagnuolo,
''Describing shapes by geometrical-topological properties of real functions''
ACM Computing Surveys, vol. 40 (2008), n. 4, 12:1–12:87.</ref>
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Size functions were introduced in
<ref name="Fro90">Patrizio Frosini, ''[http://journals.cambridge.org/download.php?file=%2FBAZ%2FBAZ42_03%2FS0004972700028574a.pdf&code=eff2726f156a5a8fdb323feb4fadd1e3 A distance for similarity classes of submanifolds of a Euclidean space]'', Bulletin of the Australian Mathematical Society, 42(3):407–416, 1990.</ref>
for the particular case of <math>M</math> equal to the topological space of all piecewise <math>C^1</math> closed paths in a <math>C^\infty</math> [[closed manifold]] embedded in a [[Euclidean space]]. Here the topology on <math>M</math> is induced by the
<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. SPIE, Intelligent Robots and Computer Vision X: Algorithms and Techniques, Boston, MA, 1607:122–133, 1991.</ref>
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>.
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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
<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
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>▼
▲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>
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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