Size function: Difference between revisions

Revision as of 22:41, 27 February 2014

Revision as of 15:05, 3 May 2015

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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>

[[File:SFesWiki.PNG|1065px|thumb|left|''An example of size function. (A) A size pair <math>(M,\varphi:M\to\mathbb{R})</math>, where <math>M</math> is the blue curve and <math>\varphi:M\to \mathbb{R}</math> is the height function. (B) The set <math>\{p\in M:\varphi(p)\le b\}</math> is depicted in green. (C) The set of points at which the [[measuring function]] <math>\varphi</math> takes a value smaller than or equal to <math>a</math>, that is, <math>\{p\in M:\varphi(p)\le a\}</math>, is depicted in red. (D) Two connected component of the set <math>\{p\in M:\varphi(p)\le b\}</math> contain at least one point in <math>\{p\in M:\varphi(p)\le a\}</math>, that is, at least one point where the [[measuring function]] <math>\varphi</math> takes a value smaller than or equal to <math>a</math>. (E) The value of the size function <math>\ell_{(M,\varphi)}</math> in the point <math>(a,b)</math> is equal to <math>2</math>.'']]

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==History and applications==

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.

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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> 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'',

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Journal of Mathematical Imaging and Vision, 21(2):107–118, 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'' 68:451–471, 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>.