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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.

[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.

[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.

[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.

[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.

[6]

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

[7]

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

[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.

[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.

[10]

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

[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.

[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.

[13]

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

[14]

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

[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.

[16]

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

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.

[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.

[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.

[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.

[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.

[6]

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

[7]

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

[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.

[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.

[10]

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

[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.

[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.

[13]

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

[14]

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

[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.

[16]

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

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