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Local well-posedness of the EPDiff equation: A survey
Möbius invariants in image recognition
1. | Department of Mathematics, ONAFT, Odessa, Ukraine |
2. | Department of Mathematics, University of Tromsø, Norway |
3. | Institute of Control Sciences of RAS, Moscow, Russia |
In this paper rational differential invariants are used to classify various plane shapes as well as plane domains equipped with an additional geometrical object.
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). 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. |
[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). 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. |
[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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