Journal of Mathematical Imaging and Vision

, Volume 50, Issue 1–2, pp 60–97 | Cite as

Overview of the Geometries of Shape Spaces and Diffeomorphism Groups



This article provides an overview of various notions of shape spaces, including the space of parametrized and unparametrized curves, the space of immersions, the diffeomorphism group and the space of Riemannian metrics. We discuss the Riemannian metrics that can be defined thereon, and what is known about the properties of these metrics. We put particular emphasis on the induced geodesic distance, the geodesic equation and its well-posedness, geodesic and metric completeness and properties of the curvature.


Shape space Diffeomorphism group Manifolds of mappings Landmark space Surface matching Riemannian geometry 



We would like to thank the referees for their careful reading of the article as well as the thoughtful comments, that helped us improve the exposition.


  1. 1.
    Alekseevsky, D., Kriegl, A., Losik, M., Michor, P.W.: The Riemannian geometry of orbit spaces—the metric, geodesics, and integrable systems. Publ. Math. (Debr.) 62(3–4), 247–276 (2003). Dedicated to Professor Lajos Tamássy on the occasion of his 80th birthday MATHMathSciNetGoogle Scholar
  2. 2.
    Aronszajn, N.: Theory of reproducing kernels. Trans. Am. Math. Soc. 68, 337–404 (1950) MATHMathSciNetGoogle Scholar
  3. 3.
    Bauer, M.: Almost local metrics on shape space of surfaces. PhD thesis, University of Vienna (2010) Google Scholar
  4. 4.
    Bauer, M., Bruveris, M.: A new Riemannian setting for surface registration. In: 3nd MICCAI Workshop on Mathematical Foundations of Computational Anatomy, pp. 182–194 (2011) Google Scholar
  5. 5.
    Bauer, M., Harms, P., Michor, P.W.: Sobolev metrics on shape space of surfaces. J. Geom. Mech. 3(4), 389–438 (2011) MATHMathSciNetGoogle Scholar
  6. 6.
    Bauer, M., Bruveris, M., Harms, P., Michor, P.W.: Vanishing geodesic distance for the Riemannian metric with geodesic equation the KdV-equation. Ann. Glob. Anal. Geom. 41(4), 461–472 (2012) MATHMathSciNetGoogle Scholar
  7. 7.
    Bauer, M., Bruveris, M., Marsland, S., Michor, P.W.: Constructing reparametrization invariant metrics on spaces of plane curves. arXiv:1207.5965 (2012)
  8. 8.
    Bauer, M., Bruveris, M., Michor, P.W.: Geodesic distance for right invariant Sobolev metrics of fractional order on the diffeomorphism group. II. Ann. Glob. Anal. Geom. 44(4), 361–368 (2013) MATHMathSciNetGoogle Scholar
  9. 9.
    Bauer, M., Bruveris, M., Michor, P.W.: The homogeneous Sobolev metric of order one on diffeomorphism groups on the real line. arXiv:1209.2836 (2012)
  10. 10.
    Bauer, M., Harms, P., Michor, P.W.: Almost local metrics on shape space of hypersurfaces in n-space. SIAM J. Imaging Sci. 5(1), 244–310 (2012) MATHMathSciNetGoogle Scholar
  11. 11.
    Bauer, M., Harms, P., Michor, P.W.: Curvature weighted metrics on shape space of hypersurfaces in n-space. Differ. Geom. Appl. 30(1), 33–41 (2012) MATHMathSciNetGoogle Scholar
  12. 12.
    Bauer, M., Harms, P., Michor, P.W.: Sobolev metrics on shape space, II: Weighted Sobolev metrics and almost local metrics. J. Geom. Mech. 4(4), 365–383 (2012) MATHMathSciNetGoogle Scholar
  13. 13.
    Bauer, M., Bruveris, M., Harms, P., Michor, P.W.: Geodesic distance for right invariant Sobolev metrics of fractional order on the diffeomorphism group. Ann. Glob. Anal. Geom. 44(1), 5–21 (2013) MATHMathSciNetGoogle Scholar
  14. 14.
    Bauer, M., Bruveris, M., Michor, P.W., Mumford, D.: Pulling back metrics from the manifold of all Riemannian metrics to the diffeomorphism group (2013, in preparation) Google Scholar
  15. 15.
    Bauer, M., Harms, P., Michor, P.W.: Sobolev metrics on the Riemannian manifold of all Riemannian metrics. J. Differ. Geom. 94(2), 187–208 (2013) MATHMathSciNetGoogle Scholar
  16. 16.
    Beg, M.F., Miller, M.I., Trouvé, A., Younes, L.: Computing large deformation metric mappings via geodesic flows of diffeomorphisms. Int. J. Comput. Vis. 61(2), 139–157 (2005) Google Scholar
  17. 17.
    Binz, E.: Two natural metrics and their covariant derivatives on a manifold of embeddings. Monatshefte Math. 89(4), 275–288 (1980) MATHMathSciNetGoogle Scholar
  18. 18.
    Binz, E., Fischer, H.R.: The manifold of embeddings of a closed manifold. In: Differential Geometric Methods in Mathematical Physics (Proc. Internat. Conf Tech. Univ. Clausthal). Clausthal-Zellerfeld, 1978. Lecture Notes in Phys., vol. 139, pp. 310–329. Springer, Berlin (1981). With an appendix by P. Michor Google Scholar
  19. 19.
    Bookstein, F.L.: The study of shape transformations after d’Arcy Thompson. Math. Biosci. 34, 177–219 (1976) MathSciNetGoogle Scholar
  20. 20.
    Bookstein, F.L.: Morphometric Tools for Landmark Data: Geometry and Biology. Cambridge University Press, Cambridge (1997) Google Scholar
  21. 21.
    Bronstein, A.M., Bronstein, M.M., Kimmel, R., Mahmoudi, M., Sapiro, G.: A Gromov-Hausdorff framework with diffusion geometry for topologically-robust non-rigid shape matching. Int. J. Comput. Vis. 89(2–3), 266–286 (2010) Google Scholar
  22. 22.
    Bruveris, M.: The energy functional on the Virasoro–Bott group with the L 2-metric has no local minima. Ann. Glob. Anal. Geom. 43(4), 385–395 (2013) MATHMathSciNetGoogle Scholar
  23. 23.
    Bruveris, M., Holm, D.D.: Geometry of image registration: The diffeomorphism group and momentum maps. arXiv:1306.6854 (2013)
  24. 24.
    Burgers, J.: A mathematical model illustrating the theory of turbulence. In: Advances in Applied Mechanics, vol. 1, pp. 171–199. Elsevier, Amsterdam (1948) Google Scholar
  25. 25.
    Camassa, R., Holm, D.D.: An integrable shallow water equation with peaked solitons. Phys. Rev. Lett. 71(11), 1661–1664 (1993) MATHMathSciNetGoogle Scholar
  26. 26.
    Cervera, V., Mascaró, F., Michor, P.W.: The action of the diffeomorphism group on the space of immersions. Differ. Geom. Appl. 1(4), 391–401 (1991) MATHGoogle Scholar
  27. 27.
    Charpiat, G., Keriven, R., philippe Pons, J., Faugeras, O.D.: Designing spatially coherent minimizing flows for variational problems based on active contours. In: International Conference on Computer Vision, vol. 2, pp. 1403–1408 (2005) Google Scholar
  28. 28.
    Clarke, B.: The completion of the manifold of Riemannian metrics with respect to its L 2 metric. PhD thesis, Leipzig (2009) Google Scholar
  29. 29.
    Clarke, B.: The metric geometry of the manifold of Riemannian metrics over a closed manifold. Calc. Var. Partial Differ. Equ. 39, 533–545 (2010) MATHGoogle Scholar
  30. 30.
    Clarke, B.: The Riemannian L 2 topology on the manifold of Riemannian metrics. Ann. Glob. Anal. Geom. 39(2), 131–163 (2011) MATHGoogle Scholar
  31. 31.
    Clarke, B.: The completion of the manifold of Riemannian metrics. J. Differ. Geom. 93(2), 203–268 (2013) MATHGoogle Scholar
  32. 32.
    Clarke, B.: Geodesics, distance, and the CAT(0) property for the manifold of Riemannian metrics. Math. Z. 273(1–2), 55–93 (2013) MATHMathSciNetGoogle Scholar
  33. 33.
    Clarke, B., Rubinstein, Y.A.: Conformal deformations of the Ebin metric and a generalized Calabi metric on the space of Riemannian metrics. Ann. Inst. Henri Poincaré, Anal. Non Linéaire 30(2), 251–274 (2013) MATHMathSciNetGoogle Scholar
  34. 34.
    Constantin, A., Kolev, B.: On the geometric approach to the motion of inertial mechanical systems. J. Phys. A 35(32), R51–R79 (2002) MATHMathSciNetGoogle Scholar
  35. 35.
    Constantin, A., Kolev, B.: Geodesic flow on the diffeomorphism group of the circle. Comment. Math. Helv. 78(4), 787–804 (2003) MATHMathSciNetGoogle Scholar
  36. 36.
    Constantin, P., Lax, P.D., Majda, A.: A simple one-dimensional model for the three-dimensional vorticity equation. Commun. Pure Appl. Math. 38(6), 715–724 (1985) MATHMathSciNetGoogle Scholar
  37. 37.
    Constantin, A., Kappeler, T., Kolev, B., Topalov, P.: On geodesic exponential maps of the Virasoro group. Ann. Glob. Anal. Geom. 31(2), 155–180 (2007) MATHMathSciNetGoogle Scholar
  38. 38.
    Cotter, C.J.: The variational particle-mesh method for matching curves. J. Phys. A, Math. Theor. 41(34), 344003 (2008) MathSciNetGoogle Scholar
  39. 39.
    Cotter, C.J., Clark, A., Peiró, J.: A reparameterisation based approach to geodesic constrained solvers for curve matching. Int. J. Comput. Vis. 99(1), 103–121 (2012) MATHMathSciNetGoogle Scholar
  40. 40.
    De Gregorio, S.: On a one-dimensional model for the three-dimensional vorticity equation. J. Stat. Phys. 59(5–6), 1251–1263 (1990) MATHGoogle Scholar
  41. 41.
    Delfour, M.C., Zolésio, J.-P.: Metrics, analysis, differential calculus, and optimization. In: Shapes and Geometries, 2nd edn. Advances in Design and Control, vol. 22. SIAM, Philadelphia (2011) Google Scholar
  42. 42.
    Dryden, I.L., Mardia, K.V.: Statistical Shape Analysis. Wiley Series in Probability and Statistics: Probability and Statistics. Wiley, Chichester (1998) MATHGoogle Scholar
  43. 43.
    Ebin, D.G.: The manifold of Riemannian metrics. In: Global Analysis (Proc. Sympos. Pure Math.), Berkeley, CA, 1968, vol. XV, pp. 11–40. Am. Math. Soc., Providence (1970) Google Scholar
  44. 44.
    Ebin, D.G., Marsden, J.: Groups of diffeomorphisms and the motion of an incompressible fluid. Ann. Math. (2) 92, 102–163 (1970) MATHMathSciNetGoogle Scholar
  45. 45.
    Eichhorn, J.: Global Analysis on Open Manifolds. Nova Science Publishers, New York (2007) MATHGoogle Scholar
  46. 46.
    Escher, J., Kolev, B.: Right-invariant Sobolev metrics of fractional order on the diffeomorphism group of the circle. arXiv:1202.5122v2 (2012)
  47. 47.
    Escher, J., Kolev, B., Wunsch, M.: The geometry of a vorticity model equation. Commun. Pure Appl. Anal. 11(4), 1407–1419 (2012) MATHMathSciNetGoogle Scholar
  48. 48.
    Fischer, A.E., Tromba, A.J.: On the Weil-Petersson metric on Teichmüller space. Trans. Am. Math. Soc. 284(1), 319–335 (1984) MATHMathSciNetGoogle Scholar
  49. 49.
    Freed, D.S., Groisser, D.: The basic geometry of the manifold of Riemannian metrics and of its quotient by the diffeomorphism group. Mich. Math. J. 36(3), 323–344 (1989) MATHMathSciNetGoogle Scholar
  50. 50.
    Gay-Balmaz, F.: Well-posedness of higher dimensional Camassa-Holm equations. Bull. Transilv. Univ. Braşov, Ser. III 2(51), 55–58 (2009) MathSciNetGoogle Scholar
  51. 51.
    Gay-Balmaz, F., Marsden, J.E., Ratiu, T.S.: The geometry of Teichmüller space and the Euler-Weil-Petersson equations (2013).
  52. 52.
    Gil-Medrano, O., Michor, P.W.: The Riemannian manifold of all Riemannian metrics. Q. J. Math. Oxford 2 (1991) Google Scholar
  53. 53.
    Glaunès, J., Vaillant, M., Miller, M.I.: Landmark matching via large deformation diffeomorphisms on the sphere. J. Math. Imaging Vis. 20(1–2), 179–200 (2004). Special issue on mathematics and image analysis Google Scholar
  54. 54.
    Grenander, U.: General Pattern Theory. Oxford University Press, London (1993) Google Scholar
  55. 55.
    Grenander, U., Miller, M.I.: Pattern Theory: from Representation to Inference. Oxford University Press, Oxford (2007) Google Scholar
  56. 56.
    Guieu, L., Roger, C.: Aspects Géométriques et Algébriques, Généralisations [Geometric and Algebraic Aspects, Generalizations]. L’algèbre et Le Groupe de Virasoro. Les Publications CRM, Montreal (2007). With an appendix by Vlad Sergiescu Google Scholar
  57. 57.
    Günther, A., Lamecker, H., Weiser, M.: Direct LDDMM of discrete currents with adaptive finite elements. In: 3rd MICCAI Workshop on Mathematical Foundations of Computational Anatomy, pp. 1–14 (2011) Google Scholar
  58. 58.
    Hamilton, R.S.: The inverse function theorem of Nash and Moser. Bull. Am. Math. Soc. 7(1), 65–222 (1982) MATHGoogle Scholar
  59. 59.
    Holm, D.D., Marsden, J.E.: Momentum maps and measure-valued solutions (peakons, filaments, and sheets) for the EPDiff equation. In: The Breadth of Symplectic and Poisson Geometry. Progr. Math., vol. 232, pp. 203–235. Birkhäuser, Boston (2005) Google Scholar
  60. 60.
    Hunter, J.K., Saxton, R.: Dynamics of director fields. SIAM J. Appl. Math. 51(6), 1498–1521 (1991) MATHMathSciNetGoogle Scholar
  61. 61.
    Hunter, J.K., Zheng, Y.X.: On a completely integrable nonlinear hyperbolic variational equation. Physica D 79(2–4), 361–386 (1994) MATHMathSciNetGoogle Scholar
  62. 62.
    Jermyn, I.H., Kurtek, S., Klassen, E., Srivastava, A.: Elastic shape matching of parameterized surfaces using square root normal fields. In: Proceedings of the 12th European Conference on Computer Vision (ECCV’12), vol. V, pp. 804–817. Springer, Berlin (2012) Google Scholar
  63. 63.
    Joshi, S.C., Miller, M.I.: Landmark matching via large deformation diffeomorphisms. IEEE Trans. Image Process. 9(8), 1357–1370 (2000) MATHMathSciNetGoogle Scholar
  64. 64.
    Jost, J.: Riemannian Geometry and Geometric Analysis, 5th edn. Universitext. Springer, Berlin (2008) MATHGoogle Scholar
  65. 65.
    Kainz, G.: A metric on the manifold of immersions and its Riemannian curvature. Monatshefte Math. 98(3), 211–217 (1984) MATHMathSciNetGoogle Scholar
  66. 66.
    Kato, T.: The Cauchy problem for quasi-linear symmetric hyperbolic systems. Arch. Ration. Mech. Anal. 58(3), 181–205 (1975) MATHGoogle Scholar
  67. 67.
    Kendall, D.G.: Shape manifolds, Procrustean metrics, and complex projective spaces. Bull. Lond. Math. Soc. 16(2), 81–121 (1984) MATHMathSciNetGoogle Scholar
  68. 68.
    Kendall, D.G., Barden, D., Carne, T.K., Le, H.: Shape and Shape Theory. Wiley Series in Probability and Statistics. Wiley, Chichester (1999) MATHGoogle Scholar
  69. 69.
    Khesin, B., Misiołek, G.: Euler equations on homogeneous spaces and Virasoro orbits. Adv. Math. 176(1), 116–144 (2003) MATHMathSciNetGoogle Scholar
  70. 70.
    Khesin, B., Lenells, J., Misiołek, G.: Generalized Hunter-Saxton equation and the geometry of the group of circle diffeomorphisms. Math. Ann. 342(3), 617–656 (2008) MATHMathSciNetGoogle Scholar
  71. 71.
    Khesin, B., Lenells, J., Misiołek, G., Preston, S.C.: Curvatures of Sobolev metrics on diffeomorphism groups. Pure Appl. Math. Q. 9(2), 291–332 (2013) MATHMathSciNetGoogle Scholar
  72. 72.
    Khesin, B., Lenells, J., Misiołek, G., Preston, S.C.: Geometry of Diffeomorphism Groups, Complete integrability and Geometric statistics. Geom. Funct. Anal. 23(1), 334–366 (2013) MATHMathSciNetGoogle Scholar
  73. 73.
    Kirillov, A.A., Yuriev, D.V.: Representations of the Virasoro algebra by the orbit method. J. Geom. Phys. 5, 351–363 (1988) MATHMathSciNetGoogle Scholar
  74. 74.
    Klassen, E., Srivastava, A., Mio, M., Joshi, S.: Analysis of planar shapes using geodesic paths on shape spaces. IEEE Trans. Pattern Anal. Mach. Intell. 26(3), 372–383 (2004) Google Scholar
  75. 75.
    Kouranbaeva, S.: The Camassa-Holm equation as a geodesic flow on the diffeomorphism group. J. Math. Phys. 40(2), 857–868 (1999) MATHMathSciNetGoogle Scholar
  76. 76.
    Kriegl, A., Michor, P.W.: The Convenient Setting of Global Analysis. Mathematical Surveys and Monographs, vol. 53. Am. Math. Soc., Providence (1997) MATHGoogle Scholar
  77. 77.
    Kriegl, A., Michor, P.W.: Regular infinite-dimensional Lie groups. J. Lie Theory 7(1), 61–99 (1997) MATHMathSciNetGoogle Scholar
  78. 78.
    Kushnarev, S.: Teichons: solitonlike geodesics on universal Teichmüller space. Exp. Math. 18(3), 325–336 (2009) MATHMathSciNetGoogle Scholar
  79. 79.
    Kushnarev, S., Narayan, A.: Approximating the Weil-Petersson metric geodesics on the universal Teichmüller space by singular solutions. arXiv:1208.2022 (2012)
  80. 80.
    Lenells, J.: The Hunter-Saxton equation describes the geodesic flow on a sphere. J. Geom. Phys. 57(10), 2049–2064 (2007) MATHMathSciNetGoogle Scholar
  81. 81.
    Manay, S., Cremers, D., Hong, B.-W., Yezzi, A.J., Soatto, S.: Integral invariants for shape matching. IEEE Trans. Pattern Anal. Mach. Intell. 28(10), 1602–1618 (2006) Google Scholar
  82. 82.
    McLachlan, R.I., Marsland, S.: N-particle dynamics of the Euler equations for planar diffeomorphisms. Dyn. Syst. 22(3), 269–290 (2007) MATHMathSciNetGoogle Scholar
  83. 83.
    Mémoli, F., Sapiro, G.: A theoretical and computational framework for isometry invariant recognition of point cloud data. Found. Comput. Math. 5(3), 313–347 (2005) MATHMathSciNetGoogle Scholar
  84. 84.
    Mennucci, A.C.G.: Metrics of curves in shape optimization and analysis. In: CIME Course on “Level Set and PDE Based Reconstruction Methods: Applications to Inverse Problems and Image Processing”, Cetraro, 2009 Google Scholar
  85. 85.
    Mennucci, A., Yezzi, A., Sundaramoorthi, G.: Properties of Sobolev-type metrics in the space of curves. Interfaces Free Bound. 10(4), 423–445 (2008) MATHMathSciNetGoogle Scholar
  86. 86.
    Micheli, M.: The differential geometry of landmark shape manifolds: metrics, geodesics, and curvature. PhD thesis, Brown University (2008) Google Scholar
  87. 87.
    Micheli, M., Michor, P.W., Mumford, D.: Sectional curvature in terms of the cometric, with applications to the Riemannian manifolds of landmarks. SIAM J. Imaging Sci. 5(1), 394–433 (2012) MATHMathSciNetGoogle Scholar
  88. 88.
    Micheli, M., Michor, P.W., Mumford, D.: Sobolev metrics on diffeomorphism groups and the derived geometry of spaces of submanifolds. Izv. Math. 77(3), 109–136 (2013) MathSciNetGoogle Scholar
  89. 89.
    Michor, P.: Manifolds of smooth maps. III. The principal bundle of embeddings of a noncompact smooth manifold. Cah. Topol. Géom. Différ. 21(3), 325–337 (1980) MATHMathSciNetGoogle Scholar
  90. 90.
    Michor, P.W.: Manifolds of Differentiable Mappings. Shiva Publ., Orpington (1980) MATHGoogle Scholar
  91. 91.
    Michor, P.W.: Some geometric evolution equations arising as geodesic equations on groups of diffeomorphisms including the Hamiltonian approach. In: Phase Space Analysis of Partial Differential Equations. Progr. Nonlinear Differential Equations Appl., vol. 69, pp. 133–215. Birkhäuser, Boston (2006) Google Scholar
  92. 92.
    Michor, P.W.: Topics in Differential Geometry. Graduate Studies in Mathematics, vol. 93. Am. Math. Soc., Providence (2008) MATHGoogle Scholar
  93. 93.
    Michor, P.W., Mumford, D.: Vanishing geodesic distance on spaces of submanifolds and diffeomorphisms. Doc. Math. 10, 217–245 (2005) (electronic) MATHMathSciNetGoogle Scholar
  94. 94.
    Michor, P.W., Mumford, D.: Riemannian geometries on spaces of plane curves. J. Eur. Math. Soc. 8, 1–48 (2006) MATHMathSciNetGoogle Scholar
  95. 95.
    Michor, P.W., Mumford, D.: An overview of the Riemannian metrics on spaces of curves using the Hamiltonian approach. Appl. Comput. Harmon. Anal. 23(1), 74–113 (2007) MATHMathSciNetGoogle Scholar
  96. 96.
    Michor, P.W., Mumford, D.: A zoo of diffeomorphism groups on \(\mathbb{R}^{n}\). Ann. Glob. Anal. Geom. 44(4), 529–540 (2013) MATHMathSciNetGoogle Scholar
  97. 97.
    Miller, M.I., Younes, L.: Group actions, homeomorphisms, and matching: a general framework. Int. J. Comput. Vis. 41, 61–84 (2001) MATHGoogle Scholar
  98. 98.
    Miller, M.I., Trouve, A., Younes, L.: On the metrics and Euler-Lagrange equations of computational anatomy. Annu. Rev. Biomed. Eng. 4, 375–405 (2002) Google Scholar
  99. 99.
    Mio, W., Srivastava, A.: Elastic-string models for representation and analysis of planar shapes. In: Proceedings of the 2004 IEEE Computer Society Conference on Computer Vision and Pattern Recognition (CVPR 2004), vol. 2, pp. II-10–II-15 (2004) Google Scholar
  100. 100.
    Mio, W., Srivastava, A., Joshi, S.: On shape of plane elastic curves. Int. J. Comput. Vis. 73(3), 307–324 (2007) Google Scholar
  101. 101.
    Misiołek, G.: A shallow water equation as a geodesic flow on the Bott-Virasoro group. J. Geom. Phys. 24(3), 203–208 (1998) MATHMathSciNetGoogle Scholar
  102. 102.
    Modin, K.: Generalised Hunter-Saxton equations and optimal information transport. arXiv:1203.4463 (2012)
  103. 103.
    Mumford, D.: Mathematical theories of shape: do they model perception? In: San Diego ’91, pp. 2–10. International Society for Optics and Photonics, San Diego (1991) Google Scholar
  104. 104.
    Mumford, D., Desolneux, A.: Pattern Theory: the Stochastic Analysis of Real-World Signals. AK Peters, Wellesley (2010) Google Scholar
  105. 105.
    Mumford, D., Michor, P.W.: On Euler’s equation and ‘EPDiff’. J. Geom. Mech. 5(3), 319–344 (2013) MATHMathSciNetGoogle Scholar
  106. 106.
    Neeb, K.-H.: Towards a Lie theory of locally convex groups. Jpn. J. Math. 1(2), 291–468 (2006) MATHMathSciNetGoogle Scholar
  107. 107.
    Okamoto, H., Sakajo, T., Wunsch, M.: On a generalization of the Constantin-Lax-Majda equation. Nonlinearity 21(10), 2447–2461 (2008) MATHMathSciNetGoogle Scholar
  108. 108.
    Ovsienko, V.Y., Khesin, B.A.: Korteweg–de Vries superequations as an Euler equation. Funct. Anal. Appl. 21, 329–331 (1987) MATHMathSciNetGoogle Scholar
  109. 109.
    Preston, S.C.: The motion of whips and chains. J. Differ. Equ. 251(3), 504–550 (2011) MATHMathSciNetGoogle Scholar
  110. 110.
    Preston, S.C.: The geometry of whips. Ann. Glob. Anal. Geom. 41(3), 281–305 (2012) MATHMathSciNetGoogle Scholar
  111. 111.
    Rumpf, M., Wirth, B.: Variational time discretization of geodesic calculus. arXiv:1210.0822 (2012)
  112. 112.
    Saitoh, S.: Theory of Reproducing Kernels and Its Applications. Pitman Research Notes in Mathematics (1988) MATHGoogle Scholar
  113. 113.
    Samir, C., Absil, P.-A., Srivastava, A., Klassen, E.: A gradient-descent method for curve fitting on Riemannian manifolds. Found. Comput. Math. 12(1), 49–73 (2012) MATHMathSciNetGoogle Scholar
  114. 114.
    Segal, G.: The geometry of the KdV equation. Int. J. Mod. Phys. A 6(16), 2859–2869 (1991) MATHGoogle Scholar
  115. 115.
    Shah, J.: H 0-type Riemannian metrics on the space of planar curves. Q. Appl. Math. 66(1), 123–137 (2008) MATHGoogle Scholar
  116. 116.
    Shah, J.: An H 2 Riemannian metric on the space of planar curves modulo similitudes (2010).
  117. 117.
    Sharon, E., Mumford, D.: 2d-shape analysis using conformal mapping. In: Proc. IEEE Conf. Computer Vision and Pattern Recognition, pp. 350–357 (2004) Google Scholar
  118. 118.
    Sharon, E., Mumford, D.: 2d-shape analysis using conformal mapping. Int. J. Comput. Vis. 70, 55–75 (2006) Google Scholar
  119. 119.
    Small, C.G.: The Statistical Theory of Shape. Springer Series in Statistics. Springer, New York (1996) MATHGoogle Scholar
  120. 120.
    Smolentsev, N.K.: Diffeomorphism groups of compact manifolds. Sovr. Mat. Prilozh. Geom. 37, 3–100 (2006) Google Scholar
  121. 121.
    Srivastava, A., Klassen, E., Joshi, S., Jermyn, I.: Shape analysis of elastic curves in Euclidean spaces. IEEE Trans. Pattern Anal. Mach. Intell. 33(7), 1415–1428 (2011) Google Scholar
  122. 122.
    Stanhope, E., Uribe, A.: The spectral function of a Riemannian orbifold. Ann. Glob. Anal. Geom. 40(1), 47–65 (2011) MATHMathSciNetGoogle Scholar
  123. 123.
    Sundaramoorthi, G., Yezzi, A., Mennucci, A.C.: Sobolev active contours. Int. J. Comput. Vis. 73(3), 345–366 (2007) Google Scholar
  124. 124.
    Sundaramoorthi, G., Yezzi, A., Mennucci, A.: Coarse-to-fine segmentation and tracking using Sobolev active contours. IEEE Trans. Pattern Anal. Mach. Intell. 30(5), 851–864 (2008) Google Scholar
  125. 125.
    Sundaramoorthi, G., Mennucci, A., Soatto, S., Yezzi, A.: A new geometric metric in the space of curves, and applications to tracking deforming objects by prediction and filtering. SIAM J. Imaging Sci. 4(1), 109–145 (2011) MATHMathSciNetGoogle Scholar
  126. 126.
    Taylor, M.E.: Partial Differential Equations I. Basic Theory, 2nd edn. Applied Mathematical Sciences, vol. 115. Springer, New York (2011) MATHGoogle Scholar
  127. 127.
    Triebel, H.: Theory of Function Spaces. II. Monographs in Mathematics, vol. 84. Birkhäuser, Basel (1992) MATHGoogle Scholar
  128. 128.
    Trouvé, A.: Diffeomorphic groups and pattern matching in image analysis. Int. J. Comput. Vis. 28, 213–221 (1998) Google Scholar
  129. 129.
    Trouvé, A., Younes, L.: Metamorphoses through Lie group action. Found. Comput. Math. 5(2), 173–198 (2005) MATHMathSciNetGoogle Scholar
  130. 130.
    Wikipedia: Smooth Riemann mapping theorem Google Scholar
  131. 131.
    Wirth, B., Bar, L., Rumpf, M., Sapiro, G.: A continuum mechanical approach to geodesics in shape space. Int. J. Comput. Vis. 93(3), 293–318 (2011) MATHMathSciNetGoogle Scholar
  132. 132.
    Wunsch, M.: On the geodesic flow on the group of diffeomorphisms of the circle with a fractional Sobolev right-invariant metric. J. Nonlinear Math. Phys. 17(1), 7–11 (2010) MATHMathSciNetGoogle Scholar
  133. 133.
    Yamada, S.: On the geometry of Weil-Petersson completion of Teichmüller spaces. Math. Res. Lett. 11(2–3), 327–344 (2004) MATHMathSciNetGoogle Scholar
  134. 134.
    Yamada, S.: Some aspects of Weil-Petersson geometry of Teichmüller spaces. In: Surveys in Geometric Analysis and Relativity. Adv. Lect. Math. (ALM), vol. 20, pp. 531–546. International Press, Somerville (2011) Google Scholar
  135. 135.
    Yezzi, A., Mennucci, A.: Metrics in the space of curves. arXiv:math/0412454 (2004)
  136. 136.
    Yezzi, A., Mennucci, A.: Conformal metrics and true “gradient flows” for curves. In: Proceedings of the Tenth IEEE International Conference on Computer Vision, vol. 1, pp. 913–919. IEEE Comput. Soc., Washington (2005) Google Scholar
  137. 137.
    Younes, L.: Computable elastic distances between shapes. SIAM J. Appl. Math. 58(2), 565–586 (1998) MATHMathSciNetGoogle Scholar
  138. 138.
    Younes, L.: Shapes and Diffeomorphisms. Springer, Berlin (2010) MATHGoogle Scholar
  139. 139.
    Younes, L., Michor, P.W., Shah, J., Mumford, D.: A metric on shape space with explicit geodesics. Atti Accad. Naz. Lincei, Rend. Lincei, Mat. Appl. 19(1), 25–57 (2008) MATHMathSciNetGoogle Scholar
  140. 140.
    Zhang, S., Younes, L., Zweck, J., Ratnanather, J.T.: Diffeomorphic surface flows: a novel method of surface evolution. SIAM J. Appl. Math. 68(3), 806–824 (2008) MATHMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  • Martin Bauer
    • 1
  • Martins Bruveris
    • 2
  • Peter W. Michor
    • 1
  1. 1.Fakultät für MathematikUniversität WienViennaAustria
  2. 2.Institut de MathématiquesEPFLLausanneSwitzerland

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