Size function: Difference between revisions

Revision as of 19:54, 30 March 2024

Revision as of 09:08, 18 July 2024

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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>

''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>

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>.