Size function#Formal definition

In size theory, the size function

$\backslash \{p\backslash in\; M:\backslash varphi(p)\backslash le\; y\backslash \}$ that contain at least one point at which the measuring function (a continuous function from a topological space $M$ to $\backslash mathbb\{R\}^k$

Patrizio Frosini and Claudia Landi, Size Theory as a Topological Tool for Computer Vision, Pattern Recognition And Image Analysis, 9(4):596–603, 1999.

Patrizio Frosini and Michele Mulazzani, Size homotopy groups for computation of natural size distances, Bulletin of the Belgian Mathematical Society, 6:455–464 1999.) $\backslash varphi$ takes a value smaller than or equal to $x$

.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.

The concept of size function can be easily extended to the case of a measuring function $\backslash varphi:M\backslash to\; \backslash mathbb\{R\}^k$, where $\backslash mathbb\{R\}^k$ is endowed with the usual partial order

.Silvia Biasotti, Andrea Cerri, Patrizio Frosini, Claudia Landi, Multidimensional size functions for shape comparison, Journal of Mathematical Imaging and Vision 32:161–179, 2008.

A survey about size functions (and size theory) can be found in.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.

File:SFesWiki.PNG $\backslash varphi$ takes a value smaller than or equal to $a$, that is, $\backslash \{p\backslash in\; M:\backslash varphi(p)\backslash le\; a\backslash \}$, is depicted in red. (D) Two connected components of the set $\backslash \{p\backslash in\; M:\backslash varphi(p)\backslash le\; b\backslash \}$ contain at least one point in $\backslash \{p\backslash in\; M:\backslash varphi(p)\backslash le\; a\backslash \}$, that is, at least one point where the measuring function $\backslash varphi$ takes a value smaller than or equal to $a$. (E) The value of the size function $\backslash ell\_\{(M,\backslash varphi)\}$ in the point $(a,b)$ is equal to $2$.]]

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Size functions were introduced in

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.

for the particular case of $M$ equal to the topological space of all piecewise $C^1$ closed paths in a $C^\backslash infty$ closed manifold embedded in a Euclidean space. Here the topology on $M$ is induced by the

$C^0$-norm, while the measuring function $\backslash varphi$ takes each path $\backslash gamma\backslash in\; M$ to its length.

In

Patrizio Frosini, Measuring shapes by size functions, Proc. SPIE, Intelligent Robots and Computer Vision X: Algorithms and Techniques, Boston, MA, 1607:122–133, 1991.

the case of $M$ equal to the topological space of all ordered $k$-tuples of points in a submanifold of a Euclidean space is considered.

Here the topology on $M$ is induced by the metric $d((P\_1,\backslash ldots,P\_k),(Q\_1\backslash ldots,Q\_k))=\backslash max\_\{1\backslash le\; i\backslash le\; k\}\backslash |P\_i-Q\_i\backslash |$.

An extension of the concept of size function to algebraic topology was made in

where the concept of size homotopy group was introduced. Here measuring functions taking values in $\backslash mathbb\{R\}^k$ are allowed.

An extension to homology theory (the size functor) was introduced in

.Francesca Cagliari, Massimo Ferri and Paola Pozzi, Size functions from a categorical viewpoint, Acta Applicandae Mathematicae, 67(3):225–235, 2001.

The concepts of size homotopy group and size functor are strictly related to the concept of persistent homology group

Herbert Edelsbrunner, David Letscher and Afra Zomorodian, Topological Persistence and Simplification, Discrete and Computational Geometry, 28(4):511–533, 2002.

studied in persistent homology. It is worth to point out that the size function is the rank of the $0$-th persistent homology group, while the relation between the persistent homology group

and the size homotopy group is analogous to the one existing between homology groups and homotopy groups.

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.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.Alessandro Verri, Claudio Uras, Patrizio Frosini and Massimo Ferri, On the use of size functions for shape analysis, Biological Cybernetics, 70:99–107, 1993.Patrizio Frosini and Claudia Landi, Size functions and morphological transformations, Acta Applicandae Mathematicae, 49(1):85–104, 1997.Alessandro Verri and Claudio Uras, Metric-topological approach to shape representation and recognition, Image Vision Comput., 14:189–207, 1996.Alessandro Verri and Claudio Uras, Computing size functions from edge maps, Internat. J. Comput. Vision, 23(2):169–183, 1997.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.Andrea Cerri, Massimo Ferri, Daniela Giorgi, Retrieval of trademark images by means of size functions Graphical Models 68:451–471, 2006.Silvia Biasotti, Daniela Giorgi, Michela Spagnuolo, Bianca Falcidieno, Size functions for comparing 3D models Pattern Recognition 41:2855–2873, 2008.

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 $\backslash Delta^+$.

Assume that $M$ is a compact locally connected Hausdorff space. The following statements hold:

• every size function $\backslash ell\_\{(M,\backslash varphi)\}(x,y)$ is a non-decreasing function in the variable $x$ and a non-increasing function in the variable $y$.
• every size function $\backslash ell\_\{(M,\backslash varphi)\}(x,y)$ is locally right-constant in both its variables.
• for every
• for every and every $y>x$, $\backslash ell\_\{(M,\backslash varphi)\}(x,y)=0$.
• for every $y\backslash ge\backslash max\; \backslash varphi$ and every

If we also assume that $M$ is a smooth closed manifold and $\backslash varphi$ is a $C^1$-function, the following useful property holds:

• in order that $(x,y)$ is a discontinuity point for $\backslash ell\_\{(M,\backslash varphi)\}$ it is necessary that either $x$ or $y$ or both are critical values for $\backslash varphi$.Patrizio Frosini, Connections between size functions and critical points, Mathematical Methods in the Applied Sciences, 19:555–569, 1996.

A strong link between the concept of size function and the concept of natural pseudodistance

$d((M,\backslash varphi),(N,\backslash psi))$ between the size pairs $(M,\backslash varphi),\backslash \; (N,\backslash psi)$ exists.Pietro Donatini and Patrizio Frosini, Lower bounds for natural pseudodistances via size functions, Archives of Inequalities and Applications, 2(1):1–12, 2004.

• if $\backslash ell\_\{(N,\backslash psi)\}(\backslash bar\; x,\backslash bar\; y)>\backslash ell\_\{(M,\backslash varphi)\}(\backslash tilde\; x,\backslash tilde\; y)$ then $d((M,\backslash varphi),(N,\backslash psi))\backslash ge\; \backslash min\backslash \{\backslash tilde\; x-\backslash bar\; x,\backslash bar\; y-\backslash tilde\; y\backslash \}$.

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.

An algebraic representation of size

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

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

Claudia Landi and Patrizio Frosini, New pseudodistances for the size function space, Proc. SPIE Vol. 3168, pp. 52–60, Vision Geometry VI, Robert A. Melter, Angela Y. Wu, Longin J. Latecki (eds.), 1997.

.Patrizio Frosini and Claudia Landi, Size functions and formal series, Appl. Algebra Engrg. Comm. Comput., 12:327–349, 2001.

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

discontinuities of the corresponding size functions, while

their multiplicities contain the information about the values taken by the

size function.

Formally:

• cornerpoints are defined as those points $p=(x,y)$, with

::$\backslash mu\; (p)\{\backslash stackrel\{\{\backslash rm\; def\}\}\{=\}\}\backslash min\; \_\{\backslash alpha\; >0\; ,\backslash beta>0\}\; \backslash ell\; \_\{(\{M\},\backslash varphi\; )\}(x+\backslash alpha\; ,y-$

\beta)-\ell _{({ M},\varphi )} (x+\alpha ,y+\beta )-

\ell_{({ M},\varphi )} (x-\alpha ,y-\beta )+\ell _{({ M}

,\varphi )} (x-\alpha ,y+\beta )

:is positive. The number $\backslash mu\; (p)$ is said to be the multiplicity of $p$.

• cornerlines and are defined as those lines $r:x=k$ such that

::

)}(k+\alpha ,y)-

\ell _{({ M},\varphi )}(k-\alpha ,y)>0.

: The number $\backslash mu\; (r)$ is sad to be the multiplicity of $r$.

• Representation Theorem: For every , it holds

::$\backslash ell\; \_\{(\{M\},\backslash varphi\; )\}(\{\backslash bar\; x\},\{\backslash bar\; y\})=\backslash sum\; \_\{p=(x,y)\backslash atop\; x\backslash le\; \{\backslash bar\; x\},\; y>\backslash bar\; y\; \}\backslash mu\backslash big(p\backslash big)+\backslash sum\; \_\{r:x=k\backslash atop\; k\backslash le\; \{\backslash bar\; x\}\; \}\backslash mu\backslash big(r\backslash big)$.

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.

{{reflist}}

• Size theory
• Natural pseudodistance
• Size functor
• Size homotopy group
• Size pair
• Matching distance
• Topological data analysis

Category:Topology

Category:Algebraic topology