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Quotient elastic metrics on the manifold of arc-length parameterized plane curves. (English) Zbl 1365.53002

This paper is devoted to the study of the pull-back of 2-parameter families of quotient metrics introduced in [“On shape of plane elastic curves”, Int. J. Comput. Vision 73, No. 3, 307–324 (2007; doi:10.1007/s11263-006-9968-0)] by W. Mio et al. The authors of the paper under review study the problem of finding geodesics between two given arc-length parameterized curves under these quotient elastic metrics, and they give a precise computation of the gradient of the energy functional in the smooth case as well as a discretization of it. Also, they implement a path-straightening method. In conclusion, the paper is interesting and worth to study by those interested in the study of elastic metrics.

MSC:

53A04 Curves in Euclidean and related spaces
58B20 Riemannian, Finsler and other geometric structures on infinite-dimensional manifolds
49Q20 Variational problems in a geometric measure-theoretic setting

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References:

[1] M. Bauer, Constructing reparametrization invariant metrics on spaces of plane curves,, Differential Geometry and its Applications, 34, 139 (2014) · Zbl 1291.58002 · doi:10.1016/j.difgeo.2014.04.008
[2] M. Bauer, Overview of the geometries of shape spaces and diffeomorphism groups,, Journal of Mathematical Imaging and Vision, 50, 60 (2014) · Zbl 1310.58005 · doi:10.1007/s10851-013-0490-z
[3] H. Brezis, <em>Functional Analysis, Sobolev Spaces and Partial Differential Equations</em>,, Springer (2011) · Zbl 1220.46002
[4] M. Bruveris, Optimal reparametrizations in the square root velocity framework,, SIAM Journal on Mathematical Analysis, 48, 4335 (2016) · Zbl 1357.58005 · doi:10.1137/15M1014693
[5] J. Brylinski, <em>Loop Spaces, Characteristic Classes and Geometric Quantization</em>,, Birkhäuser Boston (2008) · Zbl 1136.55001 · doi:10.1007/978-0-8176-4731-5
[6] V. Cervera, The action of the diffeomorphism group on the space of immersions,, Differential Geometry and its Applications, 1, 391 (1991) · Zbl 0783.58012 · doi:10.1016/0926-2245(91)90015-2
[7] T. Diez, Slice theorem for Fréchet group actions and covariant symplectic field theory, 2014,, <a href=
[8] F. Dubeau, A remark on cyclic tridiagonal matrices,, Zastosowania Matematyki Applicationes Mathematicae, 21, 253 (1991) · Zbl 0748.65022
[9] R. S. Hamilton, The inverse function theorem of Nash and Moser,, Bulletin (New series) of the American Mathematical Society, 7, 65 (1982) · Zbl 0499.58003 · doi:10.1090/S0273-0979-1982-15004-2
[10] N. J. Higham, <em>Accuracy and Stability in Numerical Algorithms: Second Edition</em>,, SIAM (2002) · Zbl 1011.65010 · doi:10.1137/1.9780898718027
[11] E. Klassen, Analysis of planar shapes using geodesic paths on shape spaces,, IEEE Transactions on Pattern Analysis and Machine Intelligence, 26, 372 (2004) · doi:10.1109/TPAMI.2004.1262333
[12] S. Lahiri, Precise matching of \(PL\) curves in \(\mathbbR^n\) in the square root velocity framework,, Geom. Imaging Comput., 2, 133 (2015) · Zbl 1403.94020 · doi:10.4310/GIC.2015.v2.n3.a1
[13] W. Mio, On shape of plane elastic curves,, International Journal of Computer Vision, 73, 307 (2007) · Zbl 1477.68398 · doi:10.1007/s11263-006-9968-0
[14] S. C. Preston, The geometry of whips,, Annals of Global Analysis Geometry, 41, 281 (2012) · Zbl 1237.58014 · doi:10.1007/s10455-011-9283-z
[15] A. Srivastava, Shape analysis of elastic curves in Euclidean spaces,, IEEE Trans. PAMI, 33, 1415 (2011) · doi:10.1109/TPAMI.2010.184
[16] C. Temperton, Algorithms for the solution of cyclic tridiagonal systems,, Journal of Computational Physics, 19, 317 (1975) · Zbl 0319.65024 · doi:10.1016/0021-9991(75)90081-9
[17] A. B. Tumpach, Gauge invariant framework for shape analysis of surfaces,, IEEE Trans Pattern Anal Mach Intell., 38, 46 (2016) · doi:10.1109/TPAMI.2015.2430319
[18] L. Younes, A metric on shape space with explicit geodesics,, Matematica e Applicazioni, 19, 25 (2008) · Zbl 1142.58013 · doi:10.4171/RLM/506
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