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June  2017, 9(2): 191-206. doi: 10.3934/jgm.2017008

Möbius invariants in image recognition

Konovenko Nadiia 1, and  Lychagin Valentin 2,3,

1. 

Department of Mathematics, ONAFT, Odessa, Ukraine
2. 

Department of Mathematics, University of Tromsø, Norway
3. 

Institute of Control Sciences of RAS, Moscow, Russia

Received  February 2016 Revised  August 2016 Published  May 2017

In this paper rational differential invariants are used to classify various plane shapes as well as plane domains equipped with an additional geometrical object.

Keywords: Lobachevskian geometry, Möbius transformations, jet-spaces, rational differential invariant, shape spaces, fingerprints.
Mathematics Subject Classification: Primary: 53A55.
Citation: Konovenko Nadiia, Lychagin Valentin. Möbius invariants in image recognition. Journal of Geometric Mechanics, 2017, 9 (2) : 191-206. doi: 10.3934/jgm.2017008
References:
[1]

P. Bibikov and V. Lychagin, Projective classification of binary and ternary forms, Journal of Geometry and Physics, 61 (2011), 1914-1927.  doi: 10.1016/j.geomphys.2011.05.001.  Google Scholar

[2]

M. Gordina and P. Lescot, Riemannian geometry of $\mathbf{Diff}(S^{1})$, Journal of Functional Analysis, 239 (2006), 611-630.  doi: 10.1016/j.jfa.2006.02.005.  Google Scholar

[3]

E. Kamke, Differentialgleichungen. Lösungsmethoden und Lösungen. Teil Ⅰ: Gewöhnliche Differentialgleichungen, Mathematik und ihre Anwendungen in Physik und Technik, Reihe A, , Bd. Leipzig: Akademische Verlagsgesellschaft, Geest and Portig K. -G., 18 (1959), xxvi+666 pp.  Google Scholar

[4]

A. A. Kirillov, Geometric approach to discrete series of unireps for Vir, J. Math. Pures Appl., 77 (1998), 735-746.  doi: 10.1016/S0021-7824(98)80007-X.  Google Scholar

[5]

N. Konovenko and V. Lychagin, Invariants of projective actions and their application to recognition of fingerprints, Anal. Math. Phys., 6 (2016), 95-107.  doi: 10.1007/s13324-015-0113-5.  Google Scholar

[6]

N. Konovenko and V. Lychagin, Lobachevskian geometry in image recognition, Lobachevskii Journal of Mathematics, 36 (2015), 286-291.  doi: 10.1134/S1995080215030075.  Google Scholar

[7]

N. Konovenko, Differential invariants and $\mathfrak{sl}_{2}$-geometries, Naukova Dumka, Kiev, (2013), 188pp. (in Russian). Google Scholar

[8]

I. S. Krasilshchik, V. V. Lychagin and A. M. Vinogradov, Geometry of Jet Spaces and Nonlinear Partial Differential Equations, Advanced Studies in Contemporary Mathematics, 1. Gordon and Breach Science Publishers, New York, 1986.  Google Scholar

[9]

B. Kruglikov and V. Lychagin, Geometry of differential equations, Handbook of global analysis, Elsevier Sci. B. V., Amsterdam, 1214 (2008), 725-771. doi: 10.1016/B978-044452833-9.50015-2.  Google Scholar

[10]

B. Kruglikov and V. Lychagin, Global Lie-Tresse theorem, Selecta Math., 22 (2016), 1357-1411.  doi: 10.1007/s00029-015-0220-z.  Google Scholar

[11]

P. Malliavin, The canonic diffusion above the diffeomorphism group of the circle, C. R. Acad. Sci. Paris, 329 (1999), 325-329.  doi: 10.1016/S0764-4442(00)88575-4.  Google Scholar

[12]

P. W. Michor and D. Mumford, An overview of the Riemannian metrics on spaces of curves using the Hamiltonian approach, Appl. Comput. Harmon. Anal., 23 (2007), 74-113.  doi: 10.1016/j.acha.2006.07.004.  Google Scholar

[13]

M. Rosenlicht, A remark on quotient spaces, An. Acad. Brasil. Ci., 35 (1963), 487-489.   Google Scholar

[14]

G. Segal, The Geometry of the KdV equation, Inter. J. of Modern Physics A, 6 (1991), 2859-2869.  doi: 10.1142/S0217751X91001416.  Google Scholar

[15]

E. Sharon and D. Mumford, 2D-Shape Analysis Using Conformal Mapping, International Journal of Computer Vision, 70 (2006), 55-75.  doi: 10.1109/CVPR.2004.1315185.  Google Scholar

[16]

E. Witten, Coadjoint Orbits of the Virasoro Group, Commun. Math. Phys., 114 (1988), 1-53.  doi: 10.1007/BF01218287.  Google Scholar

show all references

References:
[1]

P. Bibikov and V. Lychagin, Projective classification of binary and ternary forms, Journal of Geometry and Physics, 61 (2011), 1914-1927.  doi: 10.1016/j.geomphys.2011.05.001.  Google Scholar

[2]

M. Gordina and P. Lescot, Riemannian geometry of $\mathbf{Diff}(S^{1})$, Journal of Functional Analysis, 239 (2006), 611-630.  doi: 10.1016/j.jfa.2006.02.005.  Google Scholar

[3]

E. Kamke, Differentialgleichungen. Lösungsmethoden und Lösungen. Teil Ⅰ: Gewöhnliche Differentialgleichungen, Mathematik und ihre Anwendungen in Physik und Technik, Reihe A, , Bd. Leipzig: Akademische Verlagsgesellschaft, Geest and Portig K. -G., 18 (1959), xxvi+666 pp.  Google Scholar

[4]

A. A. Kirillov, Geometric approach to discrete series of unireps for Vir, J. Math. Pures Appl., 77 (1998), 735-746.  doi: 10.1016/S0021-7824(98)80007-X.  Google Scholar

[5]

N. Konovenko and V. Lychagin, Invariants of projective actions and their application to recognition of fingerprints, Anal. Math. Phys., 6 (2016), 95-107.  doi: 10.1007/s13324-015-0113-5.  Google Scholar

[6]

N. Konovenko and V. Lychagin, Lobachevskian geometry in image recognition, Lobachevskii Journal of Mathematics, 36 (2015), 286-291.  doi: 10.1134/S1995080215030075.  Google Scholar

[7]

N. Konovenko, Differential invariants and $\mathfrak{sl}_{2}$-geometries, Naukova Dumka, Kiev, (2013), 188pp. (in Russian). Google Scholar

[8]

I. S. Krasilshchik, V. V. Lychagin and A. M. Vinogradov, Geometry of Jet Spaces and Nonlinear Partial Differential Equations, Advanced Studies in Contemporary Mathematics, 1. Gordon and Breach Science Publishers, New York, 1986.  Google Scholar

[9]

B. Kruglikov and V. Lychagin, Geometry of differential equations, Handbook of global analysis, Elsevier Sci. B. V., Amsterdam, 1214 (2008), 725-771. doi: 10.1016/B978-044452833-9.50015-2.  Google Scholar

[10]

B. Kruglikov and V. Lychagin, Global Lie-Tresse theorem, Selecta Math., 22 (2016), 1357-1411.  doi: 10.1007/s00029-015-0220-z.  Google Scholar

[11]

P. Malliavin, The canonic diffusion above the diffeomorphism group of the circle, C. R. Acad. Sci. Paris, 329 (1999), 325-329.  doi: 10.1016/S0764-4442(00)88575-4.  Google Scholar

[12]

P. W. Michor and D. Mumford, An overview of the Riemannian metrics on spaces of curves using the Hamiltonian approach, Appl. Comput. Harmon. Anal., 23 (2007), 74-113.  doi: 10.1016/j.acha.2006.07.004.  Google Scholar

[13]

M. Rosenlicht, A remark on quotient spaces, An. Acad. Brasil. Ci., 35 (1963), 487-489.   Google Scholar

[14]

G. Segal, The Geometry of the KdV equation, Inter. J. of Modern Physics A, 6 (1991), 2859-2869.  doi: 10.1142/S0217751X91001416.  Google Scholar

[15]

E. Sharon and D. Mumford, 2D-Shape Analysis Using Conformal Mapping, International Journal of Computer Vision, 70 (2006), 55-75.  doi: 10.1109/CVPR.2004.1315185.  Google Scholar

[16]

E. Witten, Coadjoint Orbits of the Virasoro Group, Commun. Math. Phys., 114 (1988), 1-53.  doi: 10.1007/BF01218287.  Google Scholar

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