In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, size theory studies the properties of
topological spaces endowed with
-valued
functions, with respect to the change of these functions. More formally, the subject of size theory is the study of the
natural pseudodistance In size theory, the natural pseudodistance between two size pairs (M,\varphi:M\to \mathbb)\ , (N,\psi:N\to \mathbb)\ is the value \inf_h \, \varphi-\psi\circ h\, _\infty\ , where h\ varies in the set of all homeomorphisms from the manifold M\ to ...
between
size pairs.
A survey of 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.]
History and applications
The beginning of size theory is rooted in the concept of
size function
Size functions are shape descriptors, in a geometrical/topological sense. They are functions from the half-plane x to the natural numbers, counting certain , introduced by Frosini.
[Patrizio Frosini, ''A distance for similarity classes of submanifolds of a Euclidean space'', Bulletin of the Australian Mathematical Society, 42(3):407–416, 1990.] Size functions have been initially used as a mathematical tool for shape comparison in
computer vision
Computer vision is an interdisciplinary scientific field that deals with how computers can gain high-level understanding from digital images or videos. From the perspective of engineering, it seeks to understand and automate tasks that the hum ...
and
pattern recognition.
[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.
][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.
][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.]
An extension of the concept of size function to
algebraic topology was made in the 1999 Frosini and Mulazzani paper
[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.]
where
size homotopy group
The concept of size homotopy group is analogous in size theory of the classical concept of homotopy group. In order to give its definition, let us assume that a size pair (M,\varphi) is given, where M is a closed manifold of class C^0\ and \var ...
s were introduced, together with the
natural pseudodistance In size theory, the natural pseudodistance between two size pairs (M,\varphi:M\to \mathbb)\ , (N,\psi:N\to \mathbb)\ is the value \inf_h \, \varphi-\psi\circ h\, _\infty\ , where h\ varies in the set of all homeomorphisms from the manifold M\ to ...
for
-valued functions.
An extension to
homology theory
In mathematics, homology is a general way of associating a sequence of algebraic objects, such as abelian groups or modules, with other mathematical objects such as topological spaces. Homology groups were originally defined in algebraic topolog ...
(the
size functor
Given a size pair (M,f)\ where M\ is a manifold of dimension
n\ and f\ is an arbitrary real continuous function defined
on it, the i-th size functor, with i=0,\ldots,n\ , denoted by F_i\ , is the functor in Fun(\mathrm,\mathrm)\ , where \ ...
) was introduced in 2001.
[Francesca Cagliari, Massimo Ferri and Paola Pozzi, ''Size functions from a categorical viewpoint'', Acta Applicandae Mathematicae, 67(3):225–235, 2001.]
The
size homotopy group
The concept of size homotopy group is analogous in size theory of the classical concept of homotopy group. In order to give its definition, let us assume that a size pair (M,\varphi) is given, where M is a closed manifold of class C^0\ and \var ...
and the
size functor
Given a size pair (M,f)\ where M\ is a manifold of dimension
n\ and f\ is an arbitrary real continuous function defined
on it, the i-th size functor, with i=0,\ldots,n\ , denoted by F_i\ , is the functor in Fun(\mathrm,\mathrm)\ , where \ ...
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
-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 group
In mathematics, homology is a general way of associating a sequence of algebraic objects, such as abelian groups or modules, with other mathematical objects such as topological spaces. Homology groups were originally defined in algebraic topolog ...
s and
homotopy group
In mathematics, homotopy groups are used in algebraic topology to classify topological spaces. The first and simplest homotopy group is the fundamental group, denoted \pi_1(X), which records information about loops in a space. Intuitively, homotop ...
s.
In size theory,
size function
Size functions are shape descriptors, in a geometrical/topological sense. They are functions from the half-plane x to the natural numbers, counting certain s and
size homotopy group
The concept of size homotopy group is analogous in size theory of the classical concept of homotopy group. In order to give its definition, let us assume that a size pair (M,\varphi) is given, where M is a closed manifold of class C^0\ and \var ...
s are seen as tools to compute lower bounds for the
natural pseudodistance In size theory, the natural pseudodistance between two size pairs (M,\varphi:M\to \mathbb)\ , (N,\psi:N\to \mathbb)\ is the value \inf_h \, \varphi-\psi\circ h\, _\infty\ , where h\ varies in the set of all homeomorphisms from the manifold M\ to ...
.
Actually, the following link exists between the values taken by the size functions
,
and the
natural pseudodistance In size theory, the natural pseudodistance between two size pairs (M,\varphi:M\to \mathbb)\ , (N,\psi:N\to \mathbb)\ is the value \inf_h \, \varphi-\psi\circ h\, _\infty\ , where h\ varies in the set of all homeomorphisms from the manifold M\ to ...
between the size pairs
,
[Patrizio Frosini and Claudia Landi, ''Size Theory as a Topological Tool for Computer Vision'', Pattern Recognition And Image Analysis, 9(4):596–603, 1999.][Pietro Donatini and Patrizio Frosini, ''Lower bounds for natural pseudodistances via size functions'', Archives of Inequalities and Applications, 2(1):1–12, 2004.]
:
An analogous result holds for
size homotopy group
The concept of size homotopy group is analogous in size theory of the classical concept of homotopy group. In order to give its definition, let us assume that a size pair (M,\varphi) is given, where M is a closed manifold of class C^0\ and \var ...
.
The attempt to generalize size theory and the concept of
natural pseudodistance In size theory, the natural pseudodistance between two size pairs (M,\varphi:M\to \mathbb)\ , (N,\psi:N\to \mathbb)\ is the value \inf_h \, \varphi-\psi\circ h\, _\infty\ , where h\ varies in the set of all homeomorphisms from the manifold M\ to ...
to norms that are different from the
supremum norm has led to the study of other reparametrization invariant norms.
[Patrizio Frosini, Claudia Landi: Reparametrization invariant norms. Transactions of the American Mathematical Society 361:407–452, 2009.]
See also
*
Size function
Size functions are shape descriptors, in a geometrical/topological sense. They are functions from the half-plane x to the natural numbers, counting certain
*
Natural pseudodistance In size theory, the natural pseudodistance between two size pairs (M,\varphi:M\to \mathbb)\ , (N,\psi:N\to \mathbb)\ is the value \inf_h \, \varphi-\psi\circ h\, _\infty\ , where h\ varies in the set of all homeomorphisms from the manifold M\ to ...
*
Size functor
Given a size pair (M,f)\ where M\ is a manifold of dimension
n\ and f\ is an arbitrary real continuous function defined
on it, the i-th size functor, with i=0,\ldots,n\ , denoted by F_i\ , is the functor in Fun(\mathrm,\mathrm)\ , where \ ...
*
Size homotopy group
The concept of size homotopy group is analogous in size theory of the classical concept of homotopy group. In order to give its definition, let us assume that a size pair (M,\varphi) is given, where M is a closed manifold of class C^0\ and \var ...
*
Size pair
*
Matching distance In mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in mode ...
References
{{Topology
Topology
Algebraic topology