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In [[mathematics]], '''size theory''' studies the properties of [[topological space]]s endowed with <math>\mathbb{R}^k</math>-valued [[Function (mathematics)|functions]], with respect to the change of these functions. More formally, the subject of size theory is the study of the [[natural pseudodistance]] between [[size pair]]s.
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A survey of 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&ndash;12:87.</ref>


==History and applications==
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The beginning of size theory is rooted in the concept of [[size function]], introduced by Frosini.<ref name="Fro90">Patrizio Frosini, ''A distance for similarity classes of submanifolds of a Euclidean space'', Bulletin of the Australian Mathematical Society, 42(3):407&ndash;416, 1990.</ref> Size functions have been initially used as a mathematical tool for shape comparison in [[computer vision]] and [[pattern recognition]].<ref>Alessandro Verri, Claudio Uras, Patrizio Frosini and Massimo Ferri,
''On the use of size functions for shape analysis'',
Biological Cybernetics, 70:99&ndash;107, 1993.
</ref><ref>Patrizio Frosini and Claudia Landi,
''Size functions and morphological transformations'',
Acta Applicandae Mathematicae, 49(1):85&ndash;104, 1997.
</ref><ref>Alessandro Verri and Claudio Uras,
''Metric-topological approach to shape
representation and recognition'',
Image Vision Comput., 14:189&ndash;207, 1996.
</ref><ref>Alessandro Verri and Claudio Uras,
''Computing size functions from edge maps'',
Internat. J. Comput. Vision, 23(2):169&ndash;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&ndash;118, 2004.
</ref><ref name="dAFrLa06">Michele d'Amico, Patrizio Frosini and Claudia Landi, ''Using matching distance in Size Theory: a survey'', International Journal of Imaging Systems and Technology, 16(5):154–161, 2006.
</ref><ref name="CeFeGi06">Andrea Cerri, Massimo Ferri, Daniela Giorgi: Retrieval of trademark images by means of size functions Graphical Models 68:451&ndash;471, 2006.
</ref><ref name="BiGiSp08">Silvia Biasotti, Daniela Giorgi, Michela Spagnuolo, Bianca Falcidieno: Size functions for comparing 3D models. Pattern Recognition 41:2855&ndash;2873, 2008.</ref>
 
An extension of the concept of size function to [[algebraic topology]] was made in
,<ref name="FroMu99">Patrizio Frosini and Michele Mulazzani, ''Size homotopy groups for computation of natural size distances'', Bulletin of the Belgian Mathematical Society &ndash; Simon Stevin, 6:455&ndash;464 1999.</ref>
where [[size homotopy group]]s were introduced, together with the [[natural pseudodistance]] for <math>\mathbb{R}^k</math>-valued functions.
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&ndash;235, 2001.</ref>
The [[size homotopy group]] and the [[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&ndash;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.
 
In size theory, [[size function]]s and [[size homotopy group]]s are seen as tools to compute lower bounds for the [[natural pseudodistance]].
Actually, the following link exists between the values taken by the size functions <math>\ell_{(N,\psi)}(\bar x,\bar y)</math>, <math>\ell_{(M,\varphi)}(\tilde x,\tilde y)</math> and the [[natural pseudodistance]]
<math>d((M,\varphi),(N,\psi))</math> between the size pairs <math>(M,\varphi),\  (N,\psi)</math>
,<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&ndash;603, 1999.</ref>
:<ref name="DoFro04">Pietro Donatini and Patrizio Frosini, ''Lower bounds for natural pseudodistances via size functions'', Archives of Inequalities and Applications, 2(1):1&ndash;12, 2004.</ref>
 
: <math>\text{If }\ell_{(N,\psi)}(\bar x,\bar y)>\ell_{(M,\varphi)}(\tilde x,\tilde y)\text{ then }d((M,\varphi),(N,\psi))\ge \min\{\tilde x-\bar x,\bar y-\tilde y\}.</math>
 
An analogous result holds for [[Size homotopy group]].<ref name="FroMu99"/>
 
The attempt to generalize size theory and the concept of [[natural pseudodistance]] to norms that are different from the [[supremum norm]] has led to the study of other reparametrization invariant norms.<ref name="FrLa09">Patrizio Frosini, Claudia Landi: Reparametrization invariant norms. Transactions of the American Mathematical Society 361:407&ndash;452, 2009.</ref>
 
==References==
 
{{reflist}}
 
==See also==
* [[Size function]]
* [[Natural pseudodistance]]
* [[Size functor]]
* [[Size homotopy group]]
* [[Size pair]]
* [[Matching distance]]
 
[[Category:Topology]]
[[Category:Algebraic topology]]

Latest revision as of 17:35, 29 October 2014

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