zbMATH — the first resource for mathematics

An overview of the Riemannian metrics on spaces of curves using the Hamiltonian approach. (English) Zbl 1116.58007
Summary: Here shape space is either the manifold of simple closed smooth unparameterized curves in \(\mathbb R^2\) or is the orbifold of immersions from \(S^{1}\) to \(\mathbb R^2\) modulo the group of diffeomorphisms of \(S^{1}\). We investigate several Riemannian metrics on the shape space: \(L^2\)-metrics weighted by expressions in length and curvature. These include a scale invariant metric and a Wasserstein type metric which is sandwiched between two length-weighted metrics. Sobolev metrics of order \(n\) on curves are described. Here the horizontal projection of a tangent field is given by a pseudo-differential operator. Finally, the metric induced from the Sobolev metric on the group of diffeomorphisms on \(\mathbb R^2\) is treated. Although the quotient metrics are all given by pseudo-differential operators, their inverses are given by convolution with smooth kernels. We are able to prove local existence and uniqueness of solution to the geodesic equation for both kinds of Sobolev metrics.
We are interested in all conserved quantities, so the paper starts with the Hamiltonian setting and computes conserved momenta and geodesics in general on the space of immersions. For each metric we compute the geodesic equation on the shape space. In the end we sketch in some examples the differences between these metrics.

58D10 Spaces of embeddings and immersions
Full Text: DOI arXiv
[1] Ambrosio, L.; Gigli, N.; Savaré, G., Gradient flows with metric and differentiable structures, and applications to the Wasserstein space, Atti accad. naz. lincei cl. sci. fis. mat. natur. rend. lincei (9) mat. appl., 15, 327-343, (2004) · Zbl 1162.35349
[2] Ambrosio, L.; Gigli, N.; Savaré, G., Gradient flows in metric spaces and in the space of probability measures, Lectures in mathematics, ETH Zürich, (2005), Birkhäuser Basel · Zbl 1090.35002
[3] Adams, M.; Ratiu, T.; Schmid, R., A Lie group structure for pseudo differential operators, Math. ann., 273, 529-551, (1986) · Zbl 0587.58047
[4] Arnold, V., Sur la géométrie différentielle des groupes de Lie de dimension infinie et ses applications à l’hydrodynamique des fluides parfaits, Ann. lnst. Fourier (Grenoble), 16, 319-361, (1966) · Zbl 0148.45301
[5] Beg, M.F.; Miller, M.I.; Trouve, A.; Younes, L., Computing large deformation metric mappings via geodesic flows of diffeomorphisms, Internat. J. comput. vis., 61, 2, 139-157, (2005)
[6] Benamou, J.-D.; Brenier, Y., A computational fluid mechanics solution to the Monge Kantorovich mass transfer problem, Numer. math., 84, 375-393, (2000) · Zbl 0968.76069
[7] Abramowitz, M.; Stegun, I.A., Handbook of mathematical functions with formulas, graphs, and mathematical tables, (1992), Dover Publications, Inc. New York, Reprint of the 1972 edition · Zbl 0515.33001
[8] Kodama, H.; Michor, P.W., The homotopy type of the space of degree 0 immersed curves, Rev. mat. complut., 19, 227-234, (2006) · Zbl 1104.58003
[9] Kriegl, A.; Michor, P.W., The convenient setting for global analysis, Surveys and monographs, vol. 53, (1997), Amer. Math. Soc. Providence, RI · Zbl 0889.58001
[10] Mennucci, A.; Yezzi, A., Metrics in the space of curves · Zbl 1168.58005
[11] Mennucci, A.C.G.; Yezzi, A.; Sundaramoorthi, G., Sobolev-type metrics in the space of curves · Zbl 1168.58005
[12] Michor, P.W., Some geometric evolution equations arising as geodesic equations on groups of diffeomorphism, including the Hamiltonian approach, (), 133-215 · Zbl 1221.58006
[13] Michor, P.W.; Mumford, D., Riemannian geometries on spaces of plane curves, J. eur. math. soc. (JEMS), 8, 1-48, (2006) · Zbl 1101.58005
[14] Michor, P.W.; Mumford, D., Vanishing geodesic distance on spaces of submanifolds and diffeomorphisms, Doc. math., 10, 217-245, (2005) · Zbl 1083.58010
[15] Miller, M.I.; Younes, L., Group actions, homeomorphisms, and matching: A general framework, Internat. J. comput. vis., 41, 1/2, 61-84, (2001) · Zbl 1012.68714
[16] Miller, M.I.; Trouve, A.; Younes, L., On the metrics and euler – lagrange equations of computational anatomy, Ann. rev. biomed. engrg., 4, 375-405, (2002)
[17] W. Mio, A. Srivastava, Elastic-string models for representation and analysis of planar shapes, in: Proceedings of the IEEE Computer Society International Conference on Computer Vision and Pattern Recognition (CVPR), Washington, DC, 2004
[18] W. Mio, A. Srivastava, S. Joshi, On the shape of plane elastic curves, Preprint, Florida State University (Math/Stat/CS departments), 2005
[19] Shah, J., \(H^0\)-type Riemannian metrics on the space of planar curves
[20] A. Trouvé, Infinite dimensional group actions and pattern recognition, Technical report, DMI, Ecole Normale Superieure, 1995
[21] A. Trouvé, L. Younes, Diffeomorphic matching in 1D, in: D. Vernon (Ed.), Proc. ECCV 2000, 2000
[22] A. Trouvé, L. Younes, Local analysis on a shape manifold, Preprint, Univ. de Paris 13, 2002. Available at the URL “citeseer.ist.psu.edu/608827.html”
[23] Trouvé, A.; Younes, L., Local geometry of deformable templates, SIAM J. math. anal., 37, 1, 17-59, (2005) · Zbl 1090.58008
[24] Younes, L., Computable elastic distances between shapes, SIAM J. appl. math., 58, 565-586, (1998) · Zbl 0907.68158
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.