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| '''Size functions''' are shape descriptors, in a geometrical/topological sense. They are functions from the half-plane <math>x<y\ </math> to the natural numbers, counting certain connected components of a [[topological space]]. They are used in [[pattern recognition]] and [[topology]].
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| ==Formal definition==
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| 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
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| <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 – Simon Stevin, 6:455–464 1999.</ref>) <math>\varphi</math> takes a value smaller than or equal to <math>x\ </math>
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| .<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
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| .<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>
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| A survey about size functions (and [[size theory]]) can be found in
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| .<ref name="BiDeFa08">Silvia Biasotti, Leila De Floriani, Bianca Falcidieno, Patrizio Frosini, Daniela Giorgi, Claudia Landi, Laura Papaleo, Michela Spagnuolo,
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| Describing shapes by geometrical-topological properties of real functions,
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| ACM Computing Surveys, vol. 40 (2008), n. 4, 12:1–12:87.</ref>
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| [[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==
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| Size functions were introduced in
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| <ref name="Fro90">Patrizio Frosini, ''A distance for similarity classes of submanifolds of a Euclidean space'', Bulletin of the Australian Mathematical Society, 42(3):407–416, 1990.</ref>
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| 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
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| <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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| In
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| <ref name="Fro91">Patrizio Frosini, ''Measuring shapes by size functions'', Proc. of SPIE, Intelligent Robots and Computer Vision X: Algorithms and Techniques, Boston, MA, 1607:122–133, 1991.</ref>
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| the case of <math>M\ </math> equal to the topological space of all ordered <math>k\ </math>-tuples of points in a submanifold of a Euclidean space is considered. | |
| Here the topology on <math>M\ </math> is induced by the metric <math>d((P_1,\ldots,P_k),(Q_1\ldots,Q_k))=\max_{1\le i\le k}\|P_i-Q_i\|</math>.
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| An extension of the concept of size function to [[algebraic topology]] was made in
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| ,<ref name="FroMu99">Patrizio Frosini and Michele Mulazzani, ''Size homotopy groups for computation of natural size distances'', Bulletin of the Belgian Mathematical Society – Simon Stevin, 6:455–464 1999.</ref>
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| where the concept of [[size homotopy group]] was introduced. Here [[measuring function]]s taking values in <math>\mathbb{R}^k</math> are allowed.
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| An extension to [[homology theory]] (the [[size functor]]) was introduced in
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| .<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>
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| The concepts of [[size homotopy group]] and [[size functor]] are strictly related to the concept of [[persistent homology group]]
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| ,<ref name="EdLeZo02">Herbert Edelsbrunner, David Letscher and Afra Zomorodian, ''Topological Persistence and Simplification'', [[Discrete and Computational Geometry]], 28(4):511–533, 2002.</ref>
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| 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
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| and the size homotopy group is analogous to the one existing between [[homology group]]s and [[homotopy group]]s. | |
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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]]
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| ,<ref>Alessandro Verri, Claudio Uras, Patrizio Frosini and Massimo Ferri,
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| ''On the use of size functions for shape analysis'',
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| Biological Cybernetics, 70:99–107, 1993.</ref>
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| ,<ref>Patrizio Frosini and Claudia Landi,
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| ''Size functions and morphological transformations'',
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| Acta Applicandae Mathematicae, 49(1):85–104, 1997.</ref>
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| ,<ref>Alessandro Verri and Claudio Uras,
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| ''Metric-topological approach to shape
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| representation and recognition'',
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| Image Vision Comput., 14:189–207, 1996.</ref>
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| ,<ref>Alessandro Verri and Claudio Uras,
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| ''Computing size functions from edge maps'',
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| Internat. J. Comput. Vision, 23(2):169–183, 1997.</ref>
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| ,<ref>Françoise Dibos, Patrizio Frosini and Denis Pasquignon,
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| ''The use of size functions for comparison of shapes through differential invariants'', | |
| Journal of Mathematical Imaging and Vision, 21(2):107–118, 2004.</ref>
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| ,<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>
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| <ref name="BiGiSp08">Silvia Biasotti, Daniela Giorgi, Michela Spagnuolo, Bianca Falcidieno: Size functions for comparing 3D models. Pattern Recognition 41:2855–2873, 2008.</ref>
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| 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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| ==Main properties==
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| Assume that <math>M\ </math> is a compact locally connected Hausdorff space. The following statements hold:
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| ¤ every size function <math>\ell_{(M,\varphi)}(x,y)</math> is a [[non-decreasing function]] in the variable <math>x\ </math> and a [[Nonincreasing function|non-increasing function]] in the variable <math>y\ </math>.
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| ¤ every size function <math>\ell_{(M,\varphi)}(x,y)</math> is locally right-constant in both its variables.
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| ¤ for every <math>x<y\ </math>, <math>\ell_{(M,\varphi)}(x,y)</math> is finite.
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| ¤ for every <math>x<\min \varphi</math> and every <math>y>x\ </math>, <math>\ell_{(M,\varphi)}(x,y)=0</math>.
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| ¤ 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>.
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| 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:
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| ¤ 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>.
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| <ref name="Fro96">Patrizio Frosini, ''Connections between size functions and critical points'', Mathematical Methods In The Applied Sciences, 19:555–569, 1996.</ref>
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| A strong link between the concept of size function and the concept of [[natural pseudodistance]]
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| <math>d((M,\varphi),(N,\psi))</math> between the size pairs <math>(M,\varphi),\ (N,\psi)</math> exists
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| ,<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>
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| :<ref name="DoFro04">Pietro Donatini and Patrizio Frosini, ''Lower bounds for natural pseudodistances via size functions'', Archives of Inequalities and Applications, 2(1):1–12, 2004.</ref>
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| ¤ if <math>\ell_{(N,\psi)}(\bar x,\bar y)>\ell_{(M,\varphi)}(\tilde x,\tilde y)</math> then <math>d((M,\varphi),(N,\psi))\ge \min\{\tilde x-\bar x,\bar y-\tilde y\}</math>.
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| The previous result gives an easy way to get lower bounds for the [[natural pseudodistance]] and is one of the main motivation to introduce the concept of size function. | |
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| ==Representation by formal series==
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| <!-- Image with unknown copyright status removed: [[File:FSesWiki.PNG|260px|thumb|right|''An example of formal representation by formal series for a given size function. In this case <math>\mu(p_1)=1</math>, <math>\mu(p_2)=2</math>, <math>\mu(p_4)=1</math>, <math>\mu(r)=1</math>. Note that <math>\mu(p_3)=0</math>. The associated represantion by formal series is then <math>r+p_1+2p_2+p_4</math>.'']] -->
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| An algebraic representation of size
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| functions in terms of collections of points and lines in the real plane with
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| multiplicities, i.e. as particular formal series, was furnished in
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| ,<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>
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| .<ref name="FroLa01">Patrizio Frosini and Claudia Landi, ''Size functions and formal series'', Appl. Algebra Engrg. Comm. Comput., 12:327–349, 2001.</ref>
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| The points (called ''cornerpoints'') and lines (called ''cornerlines'') of such formal series encode the information about
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| discontinuities of the corresponding size functions, while
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| their multiplicities contain the information about the values taken by the
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| size function.
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| Formally:
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| * ''cornerpoints'' are defined as those points <math>p=(x,y)\ </math>, with <math>x<y\ </math>, such that the number
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| <math>\mu (p){\stackrel{{\rm def}}{=}}\min _{\alpha >0 ,\beta>0} \ell _{({M},\varphi )}(x+\alpha ,y-
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| \beta)-\ell _{({ M},\varphi )} (x+\alpha ,y+\beta )-
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| \ell_{({ M},\varphi )} (x-\alpha ,y-\beta )+\ell _{({ M}
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| ,\varphi )} (x-\alpha ,y+\beta )</math>
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| is positive. | |
| The number <math>\mu (p)\ </math> is said to be the ''multiplicity'' of <math>p\ </math>. | |
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| * ''cornerlines'' and are defined as those lines <math>r:x=k\ </math> such that
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| <math>\mu (r){\stackrel{\rm def}{=}}\min _{\alpha >0 ,k+\alpha <y}\ell _{({ M},\varphi
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| )}(k+\alpha ,y)-
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| \ell _{({ M},\varphi )}(k-\alpha ,y)>0.</math>
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| The number <math>\mu (r)\ </math> is sad to be the '' multiplicity'' of <math>r\ </math>.
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| * ''Representation Theorem'': For every <math>{\bar x}<{\bar y}</math>, it holds <math>\ell _{({M},\varphi )}({\bar x},{\bar y})=\sum _{p=(x,y)\atop x\le {\bar x}, y>\bar y }\mu\big(p\big)+\sum _{r:x=k\atop k\le {\bar x} }\mu\big(r\big)</math>
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| This representation contains the
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| same amount of information about the shape under study as the original
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| size function does, but is much more concise.
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| This algebraic approach to size functions leads to the definition of new similarity measures
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| between shapes, by translating the problem of comparing size functions into
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| 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>
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| ==References==
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| {{reflist}}
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| ==See also==
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| * [[Size theory]]
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| * [[Natural pseudodistance]]
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| * [[Size functor]]
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| * [[Size homotopy group]]
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| * [[Size pair]]
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| * [[Matching distance]]
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| * [[Topological data analysis]]
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| [[Category:Topology]]
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| [[Category:Algebraic topology]]
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