Size function: Difference between revisions

Revision as of 15:20, 3 May 2015

Revision as of 05:55, 4 May 2015

Line 5:

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 [[measuring function]] (a [[continuous function]] from a [[topological space]] <math>M\ </math> to <math>\mathbb{R}^k\ </math>

.<ref name="FroLa99">Patrizio Frosini, 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, 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>

.<ref name="dAFrLa06">Michele d'Amico, Patrizio Frosini, Claudia Landi, ''Using matching distance in Size Theory: a survey'', International Journal of Imaging Systems and Technology, 16(5):154–161, 2006.</ref>. 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>.

[[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>.'']]

Line 28:

where the concept of [[size homotopy group]] was introduced. Here [[measuring function]]s taking values in <math>\mathbb{R}^k</math> are allowed.

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 [[persistent homology group]]

<ref name="EdLeZo02">Herbert Edelsbrunner, David Letscher and Afra Zomorodian, ''Topological Persistence and Simplification'', [[Discrete and Computational Geometry]], 28(4):511–533, 2002.</ref>

Line 35:

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">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> 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'',

Line 52 ⟶ 53:

''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="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>.

Line 70:

¤ for every <math>y\ge\max \varphi</math> and every <math>x<y\ </math>, <math>\ell_{(M,\varphi)}(x,y)</math> equals the number of connected components of <math>M\ </math> on which the minimum value of <math>\varphi</math> is smaller than or equal to <math>x\ </math>.

If we also assume that <math>M\ </math> is a smooth [[closed manifold]] and <math>\varphi</math> is a <math>C^1\ </math>-function, the following useful property holds:

¤ in order that <math>(x,y)\ </math> is a discontinuity point for <math>\ell_{(M,\varphi)}</math> it is necessary that either <math>x\ </math> or <math>y\ </math> or both are critical values for <math>\varphi</math>.

Line 89:

functions in terms of collections of points and lines in the real plane with

multiplicities, i.e. as particular formal series, was furnished in

<ref name="LaFro97">Claudia Landi and Patrizio Frosini, ''New pseudodistances for the size function space'', Proc. SPIE Vol. 3168, p. 52-60, Vision Geometry VI, Robert A. Melter, Angela Y. Wu, Longin J. Latecki (eds.), 1997.</ref>

<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="FroLa01LaFro97">PatrizioClaudia FrosiniLandi and ClaudiaPatrizio LandiFrosini, ''SizeNew functionspseudodistances andfor the size formalfunction seriesspace'', ApplProc. AlgebraSPIE EngrgVol. Comm3168, p. Comput52-60, Vision Geometry VI, Robert A. Melter, 12:327–349Angela Y. Wu, 2001Longin J. Latecki (eds.), 1997.</ref>.

.<ref name="FroLa01">Patrizio Frosini and Claudia Landi, ''Size functions and formal series'', Appl. Algebra Engrg. Comm. Comput., 12:327–349, 2001.</ref>

The points (called ''cornerpoints'') and lines (called ''cornerlines'') of such formal series encode the information about

discontinuities of the corresponding size functions, while

Line 117:

This representation contains the

same amount of information about the shape under study as the original

size function does, but is much more concise.

This algebraic approach to size functions leads to the definition of new similarity measures

between shapes, by translating the problem of comparing size functions into

the problem of comparing formal series. The most studied among these metrics between size function is the [[matching distance]].<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>.

==References==