Size functions are shape descriptors, in a geometrical/topological sense. They are functions from the half-plane to the natural numbers, counting certain connected components of a topological space. They are used in pattern recognition and topology.

Formal definition

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In size theory, the size function   associated with the size pair   is defined in the following way. For every  ,   is equal to the number of connected components of the set   that contain at least one point at which the measuring function (a continuous function from a topological space   to   [1] [2])   takes a value smaller than or equal to   .[3] The concept of size function can be easily extended to the case of a measuring function  , where   is endowed with the usual partial order .[4] A survey about size functions (and size theory) can be found in.[5]

 
An example of size function. (A) A size pair  , where   is the blue curve and   is the height function. (B) The set   is depicted in green. (C) The set of points at which the measuring function   takes a value smaller than or equal to  , that is,  , is depicted in red. (D) Two connected components of the set   contain at least one point in  , that is, at least one point where the measuring function   takes a value smaller than or equal to  . (E) The value of the size function   in the point   is equal to  .

History and applications

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Size functions were introduced in [6] for the particular case of   equal to the topological space of all piecewise   closed paths in a   closed manifold embedded in a Euclidean space. Here the topology on   is induced by the  -norm, while the measuring function   takes each path   to its length. In [7] the case of   equal to the topological space of all ordered  -tuples of points in a submanifold of a Euclidean space is considered. Here the topology on   is induced by the metric  .

An extension of the concept of size function to algebraic topology was made in [2] where the concept of size homotopy group was introduced. Here measuring functions taking values in   are allowed. An extension to homology theory (the size functor) was introduced in .[8] The concepts of size homotopy group and size functor are strictly related to the concept of persistent homology group [9] 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 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.[3][10][11][12][13][14][15][16][17] 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  .

Main properties

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

  • every size function   is a non-decreasing function in the variable   and a non-increasing function in the variable  .
  • every size function   is locally right-constant in both its variables.
  • for every  ,   is finite.
  • for every   and every  ,  .
  • for every   and every  ,   equals the number of connected components of   on which the minimum value of   is smaller than or equal to  .

If we also assume that   is a smooth closed manifold and   is a  -function, the following useful property holds:

  • in order that   is a discontinuity point for   it is necessary that either   or   or both are critical values for  .[18]

A strong link between the concept of size function and the concept of natural pseudodistance   between the size pairs   exists.[1][19]

  • if   then  .

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.

Representation by formal series

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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 [1] [20] .[21] 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  , with  , such that the number
 
is positive. The number   is said to be the multiplicity of  .
  • cornerlines and are defined as those lines   such that
 
The number   is sad to be the multiplicity of  .
  • Representation Theorem: For every  , it holds
 .

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.[3]

References

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  1. ^ a b c Patrizio Frosini and Claudia Landi, Size Theory as a Topological Tool for Computer Vision, Pattern Recognition And Image Analysis, 9(4):596–603, 1999.
  2. ^ a b Patrizio Frosini and Michele Mulazzani, Size homotopy groups for computation of natural size distances, Bulletin of the Belgian Mathematical Society, 6:455–464 1999.
  3. ^ a b c 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.
  4. ^ Silvia Biasotti, Andrea Cerri, Patrizio Frosini, Claudia Landi, Multidimensional size functions for shape comparison, Journal of Mathematical Imaging and Vision 32:161–179, 2008.
  5. ^ 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.
  6. ^ Patrizio Frosini, A distance for similarity classes of submanifolds of a Euclidean space, Bulletin of the Australian Mathematical Society, 42(3):407–416, 1990.
  7. ^ Patrizio Frosini, Measuring shapes by size functions, Proc. SPIE, Intelligent Robots and Computer Vision X: Algorithms and Techniques, Boston, MA, 1607:122–133, 1991.
  8. ^ Francesca Cagliari, Massimo Ferri and Paola Pozzi, Size functions from a categorical viewpoint, Acta Applicandae Mathematicae, 67(3):225–235, 2001.
  9. ^ Herbert Edelsbrunner, David Letscher and Afra Zomorodian, Topological Persistence and Simplification, Discrete and Computational Geometry, 28(4):511–533, 2002.
  10. ^ Claudio Uras and Alessandro Verri, Describing and recognising shape through size functions ICSI Technical Report TR-92-057, Berkeley, 1992.
  11. ^ Alessandro Verri, Claudio Uras, Patrizio Frosini and Massimo Ferri, On the use of size functions for shape analysis, Biological Cybernetics, 70:99–107, 1993.
  12. ^ Patrizio Frosini and Claudia Landi, Size functions and morphological transformations, Acta Applicandae Mathematicae, 49(1):85–104, 1997.
  13. ^ Alessandro Verri and Claudio Uras, Metric-topological approach to shape representation and recognition, Image Vision Comput., 14:189–207, 1996.
  14. ^ Alessandro Verri and Claudio Uras, Computing size functions from edge maps, Internat. J. Comput. Vision, 23(2):169–183, 1997.
  15. ^ 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.
  16. ^ Andrea Cerri, Massimo Ferri, Daniela Giorgi, Retrieval of trademark images by means of size functions Graphical Models 68:451–471, 2006.
  17. ^ Silvia Biasotti, Daniela Giorgi, Michela Spagnuolo, Bianca Falcidieno, Size functions for comparing 3D models Pattern Recognition 41:2855–2873, 2008.
  18. ^ Patrizio Frosini, Connections between size functions and critical points, Mathematical Methods in the Applied Sciences, 19:555–569, 1996.
  19. ^ Pietro Donatini and Patrizio Frosini, Lower bounds for natural pseudodistances via size functions, Archives of Inequalities and Applications, 2(1):1–12, 2004.
  20. ^ 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.
  21. ^ Patrizio Frosini and Claudia Landi, Size functions and formal series, Appl. Algebra Engrg. Comm. Comput., 12:327–349, 2001.

See also

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