Law of the unconscious statistician: Difference between revisions

From formulasearchengine
Jump to navigation Jump to search
Newer edit →
en>Khazar2
clean up, typos fixed: well-known → well known using AWB (8097)
 
Uniformization of notation according to the article "Expected value"
Line 1: Line 1:
In [[mathematics]], '''size theory''' studies the properties of [[topological space]]s endowed with <math>\mathbb{R}^k</math>-valued [[Function (mathematics)|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 pair]]s.
A survey of size theory can be found in
.<ref name="BiDeFa08">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&ndash;12:87.</ref>


==History and applications==
The beginning of size theory is rooted in the concept of [[size function]], introduced by Frosini.<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&ndash;416, 1990.</ref> Size functions have been initially used as a mathematical tool for shape comparison in [[computer vision]] and [[pattern recognition]].<ref>Alessandro Verri, Claudio Uras, Patrizio Frosini and Massimo Ferri,
''On the use of size functions for shape analysis'',
Biological Cybernetics, 70:99&ndash;107, 1993.
</ref><ref>Patrizio Frosini and Claudia Landi,
''Size functions and morphological transformations'',
Acta Applicandae Mathematicae, 49(1):85&ndash;104, 1997.
</ref><ref>Alessandro Verri and Claudio Uras,
''Metric-topological approach to shape
representation and recognition'',
Image Vision Comput., 14:189&ndash;207, 1996.
</ref><ref>Alessandro Verri and Claudio Uras,
''Computing size functions from edge maps'',
Internat. J. Comput. Vision, 23(2):169&ndash;183, 1997.
</ref><ref>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&ndash;118, 2004.
</ref><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&ndash;471, 2006.
</ref><ref name="BiGiSp08">Silvia Biasotti, Daniela Giorgi, Michela Spagnuolo, Bianca Falcidieno: Size functions for comparing 3D models. Pattern Recognition 41:2855&ndash;2873, 2008.</ref>


The kinds of kayaks utilized in kayak sportfishing commonly so much unlike common kayaks. Consequently, [http://www.youtube.com/watch?v=exmOly_qC8U Kayak paddle reviews] day fishing includes involving expansive kayaks, take a seat [http://www.youtube.com/watch?v=YEtPp9RSW4c sit on top kayak reviews] finest kayaks and also conjunction kayaks. Another particular canoe put to use solely concerning reef [http://www.youtube.com/watch?v=QrMw2niRTXg fishing kayaks reviews] is the resist angling canoe. Angling after a canoe as well has an benefit of having this avenues that could n't have stay ready commonly.   <br> There are certainly genuinely [http://www.youtube.com/watch?v=NMuXVs8MuPQ inflatable kayak reviews] kayaks obtainable. If you undertake n't have much memory, this is the boat to decide on.  gold watches It is typically flat to fill just under half of it is traditional space. The material is truly 100%  gold watches rubber vinyl. Because of this it can actually be applied in rougher seas. This type is also ideal for amateurs that go ahead and learning to [http://www.youtube.com/watch?v=dxC-dprhv98 perception kayak reviews]. <br><br> The exact functionality and comfort available from inflatable kayaks have authorized so many water sports fanatics to be able to need windsurfing. As a result, paddlers can now seek many rural channels and additionally canals without afraid about the convert concerning kayaks. Furthermore, all kayaks can be more challenging and can conveniently gliding on top of boulders or perhaps compress between them.
An extension of the concept of size function to [[algebraic topology]] was made in  
,<ref name="FroMu99">Patrizio Frosini and Michele Mulazzani, ''Size homotopy groups for computation of natural size distances'', Bulletin of the Belgian Mathematical Society &ndash; Simon Stevin, 6:455&ndash;464 1999.</ref>
where [[size homotopy group]]s were introduced, together with the [[natural pseudodistance]] for <math>\mathbb{R}^k</math>-valued functions.
An extension to [[homology theory]] (the [[size functor]]) was introduced in
.<ref name="CaFePo01">Francesca Cagliari, Massimo Ferri and Paola Pozzi, ''Size functions from a categorical viewpoint'', Acta Applicandae Mathematicae, 67(3):225&ndash;235, 2001.</ref>
The [[size homotopy group]] and the [[size functor]] are strictly related to the concept of [[persistent homology group]]
,<ref name="EdLeZo02">Herbert Edelsbrunner, David Letscher and Afra Zomorodian, ''Topological Persistence and Simplification'', [[Discrete and Computational Geometry]], 28(4):511&ndash;533, 2002.</ref>
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
and the size homotopy group is analogous to the one existing between [[homology group]]s and [[homotopy group]]s.
 
In size theory, [[size function]]s and [[size homotopy group]]s 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 <math>\ell_{(N,\psi)}(\bar x,\bar y)</math>, <math>\ell_{(M,\varphi)}(\tilde x,\tilde y)</math> and the [[natural pseudodistance]]
<math>d((M,\varphi),(N,\psi))</math> between the size pairs <math>(M,\varphi),\  (N,\psi)</math>
,<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&ndash;603, 1999.</ref>
:<ref name="DoFro04">Pietro Donatini and Patrizio Frosini, ''Lower bounds for natural pseudodistances via size functions'', Archives of Inequalities and Applications, 2(1):1&ndash;12, 2004.</ref>
 
: <math>\text{If }\ell_{(N,\psi)}(\bar x,\bar y)>\ell_{(M,\varphi)}(\tilde x,\tilde y)\text{ then }d((M,\varphi),(N,\psi))\ge \min\{\tilde x-\bar x,\bar y-\tilde y\}.</math>
 
An analogous result holds for [[Size homotopy group]].<ref name="FroMu99"/>
 
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.<ref name="FrLa09">Patrizio Frosini, Claudia Landi: Reparametrization invariant norms. Transactions of the American Mathematical Society 361:407&ndash;452, 2009.</ref>
 
==References==
 
{{reflist}}
 
==See also==
* [[Size function]]
* [[Natural pseudodistance]]
* [[Size functor]]
* [[Size homotopy group]]
* [[Size pair]]
* [[Matching distance]]
 
[[Category:Topology]]
[[Category:Algebraic topology]]

Revision as of 16:50, 9 October 2013

In 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 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 ,[11] where size homotopy groups were introduced, together with the natural pseudodistance for -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 -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 , and the natural pseudodistance between the size pairs ,[14]

[15]

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]

References

43 year old Petroleum Engineer Harry from Deep River, usually spends time with hobbies and interests like renting movies, property developers in singapore new condominium and vehicle racing. Constantly enjoys going to destinations like Camino Real de Tierra Adentro.

See also

  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. ↑ 11.0 11.1 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.
Retrieved from "https://en.formulasearchengine.com/index.php?title=Law_of_the_unconscious_statistician&oldid=23838"