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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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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>
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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