An overview of the Riemannian metrics on spaces of curves using the Hamiltonian approach

Applied and Computational Harmonic Analysis

Volumn 23, Issue 1, 2007, Pages 74-113

  Michor, Peter W ^a,b   Mumford, David ^c  

^a UNIVERSITY OF VIENNA   (Austria)

^b Erwin Schrodinger International Institute for Mathematical Physics   (Austria)

^c BROWN UNIVERSITY   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

DIFFERENTIAL EQUATIONS; HAMILTONIANS; MATHEMATICAL OPERATORS;

DIFFEOMORPHISMS; GEODESIC EQUATION; SHAPE SPACE; SOBOLEV METRICS;

SHAPE OPTIMIZATION;

EID: 34249788990     PISSN: 10635203     EISSN: 1096603X     Source Type: Journal    
DOI: 10.1016/j.acha.2006.07.004     Document Type: Article

References (24)

1. Ambrosio L., Gigli N., and 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 (2004) 327-343
2. Ambrosio L., Gigli N., and Savaré G. Gradient Flows in Metric Spaces and in the Space of Probability Measures. Lectures in Mathematics, ETH Zürich (2005), Birkhäuser, Basel
3. Adams M., Ratiu T., and Schmid R. A Lie group structure for pseudo differential operators. Math. Ann. 273 (1986) 529-551
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 (1966) 319-361
5. Beg M.F., Miller M.I., Trouve A., and Younes L. Computing large deformation metric mappings via geodesic flows of diffeomorphisms. Internat. J. Comput. Vis. 61 2 (2005) 139-157
6. Benamou J.-D., and Brenier Y. A computational fluid mechanics solution to the Monge Kantorovich mass transfer problem. Numer. Math. 84 (2000) 375-393
7. Abramowitz M., and Stegun I.A. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (1992), Dover Publications, Inc., New York Reprint of the 1972 edition
8. Kodama H., and Michor P.W. The homotopy type of the space of degree 0 immersed curves. Rev. Mat. Complut. 19 (2006) 227-234. math/0509694
9. Kriegl A., and Michor P.W. The Convenient Setting for Global Analysis. Surveys and Monographs vol. 53 (1997), Amer. Math. Soc., Providence, RI
10. Mennucci A., and Yezzi A. Metrics in the space of curves. math.DG/0412454
11. Mennucci A.C.G., Yezzi A., and Sundaramoorthi G. Sobolev-type metrics in the space of curves. math.DG/0605017
12. Michor P.W. Some geometric evolution equations arising as geodesic equations on groups of diffeomorphism, including the Hamiltonian approach. Phase Space Analysis of Partial Differential Equations. Progress in Non Linear Differential Equations and Their Applications vol. 69 (2006), Birkhäuser, Boston 133-215
13. Michor P.W., and Mumford D. Riemannian geometries on spaces of plane curves. J. Eur. Math. Soc. (JEMS) 8 (2006) 1-48. math.DG/0312384
14. Michor P.W., and Mumford D. Vanishing geodesic distance on spaces of submanifolds and diffeomorphisms. Doc. Math. 10 (2005) 217-245. math.DG/0409303
15. Miller M.I., and Younes L. Group actions, homeomorphisms, and matching: A general framework. Internat. J. Comput. Vis. 41 1/2 (2001) 61-84
16. Miller M.I., Trouve A., and Younes L. On the metrics and Euler-Lagrange equations of computational anatomy. Ann. Rev. Biomed. Engrg. 4 (2002) 375-405
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. 0-type Riemannian metrics on the space of planar curves. math.DG/0510192
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., and Younes L. Local geometry of deformable templates. SIAM J. Math. Anal. 37 1 (2005) 17-59
24. Younes L. Computable elastic distances between shapes. SIAM J. Appl. Math. 58 (1998) 565-586