Size theory

In mathematics, size theory studies the properties of topological spaces endowed with ${\displaystyle \mathbb {R} ^{k}}$-valued functions, with respect to the change of these functions. More formally, the subject of size theory is the study of the natural pseudodistance between size pairs. A survey of size theory can be found in .[1]

History and applications

The beginning of size theory is rooted in the concept of size function, introduced by Frosini.[2] Size functions have been initially used as a mathematical tool for shape comparison in computer vision and pattern recognition.[3][4][5][6][7][8][9][10]

An extension of the concept of size function to algebraic topology was made in Size homotopy groups for computation of natural size distances,[11] where size homotopy groups were introduced, together with the natural pseudodistance for ${\displaystyle \mathbb {R} ^{k}}$-valued functions. An extension to homology theory (the size functor) was introduced in .[12] The size homotopy group and the size functor are strictly related to the concept of persistent homology group ,[13] studied in persistent homology. It is worth to point out that the size function is the rank of the ${\displaystyle 0}$-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.

In size theory, size functions and size homotopy groups are seen as tools to compute lower bounds for the natural pseudodistance. Actually, the following link exists between the values taken by the size functions ${\displaystyle \ell _{(N,\psi )}({\bar {x}},{\bar {y}})}$, ${\displaystyle \ell _{(M,\varphi )}({\tilde {x}},{\tilde {y}})}$ and the natural pseudodistance ${\displaystyle d((M,\varphi ),(N,\psi ))}$ between the size pairs ${\displaystyle (M,\varphi ),\ (N,\psi )}$ ,[14]

[15]

${\displaystyle {\text{If }}\ell _{(N,\psi )}({\bar {x}},{\bar {y}})>\ell _{(M,\varphi )}({\tilde {x}},{\tilde {y}}){\text{ then }}d((M,\varphi ),(N,\psi ))\geq \min\{{\tilde {x}}-{\bar {x}},{\bar {y}}-{\tilde {y}}\}.}$

An analogous result holds for size homotopy group.[11]

The attempt to generalize size theory and the concept of natural pseudodistance to norms that are different from the supremum norm has led to the study of other reparametrization invariant norms.[16]

See also

• Size function
• Natural pseudodistance
• Size functor
• Size homotopy group
• Size pair
• Matching distance

References

1. 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.
2. Patrizio Frosini, A distance for similarity classes of submanifolds of a Euclidean space, Bulletin of the Australian Mathematical Society, 42(3):407–416, 1990.
3. Alessandro Verri, Claudio Uras, Patrizio Frosini and Massimo Ferri, On the use of size functions for shape analysis, Biological Cybernetics, 70:99–107, 1993.
4. Patrizio Frosini and Claudia Landi, Size functions and morphological transformations, Acta Applicandae Mathematicae, 49(1):85–104, 1997.
5. Alessandro Verri and Claudio Uras, Metric-topological approach to shape representation and recognition, Image Vision Comput., 14:189–207, 1996.
6. Alessandro Verri and Claudio Uras, Computing size functions from edge maps, Internat. J. Comput. Vision, 23(2):169–183, 1997.
7. 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.
8. 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.
9. Andrea Cerri, Massimo Ferri, Daniela Giorgi: Retrieval of trademark images by means of size functions Graphical Models 68:451–471, 2006.
10. Silvia Biasotti, Daniela Giorgi, Michela Spagnuolo, Bianca Falcidieno: Size functions for comparing 3D models. Pattern Recognition 41:2855–2873, 2008.
11. 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.
12. Francesca Cagliari, Massimo Ferri and Paola Pozzi, Size functions from a categorical viewpoint, Acta Applicandae Mathematicae, 67(3):225–235, 2001.
13. Herbert Edelsbrunner, David Letscher and Afra Zomorodian, Topological Persistence and Simplification, Discrete and Computational Geometry, 28(4):511–533, 2002.
14. Patrizio Frosini and Claudia Landi, Size Theory as a Topological Tool for Computer Vision, Pattern Recognition And Image Analysis, 9(4):596–603, 1999.
15. Pietro Donatini and Patrizio Frosini, Lower bounds for natural pseudodistances via size functions, Archives of Inequalities and Applications, 2(1):1–12, 2004.
16. Patrizio Frosini, Claudia Landi: Reparametrization invariant norms. Transactions of the American Mathematical Society 361:407–452, 2009.