Abstract
This chapter reviews some past and recent developments in shape comparison and analysis of curves based on the computation of intrinsic Riemannian metrics on the space of curve modulo shape-preserving transformations. We summarize the general construction and theoretical properties of quotient elastic metrics for Euclidean as well as non-Euclidean curves before considering the special case of the square root velocity metric for which the expression of the resulting distance simplifies through a particular transformation. We then examine the different numerical approaches that have been proposed to estimate such distances in practice and in particular to quotient out curve reparametrization in the resulting minimization problems.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
To be mathematically exact, one should limit oneself to the slightly smaller set of free immersions in this definition,as the quotient space has some mild singularities without this restriction. We will, however, ignore this subtlety for thepurpose of this book chapter.
References
Bauer, M., Harms, P., Michor, P.W.: Sobolev metrics on shape space of surfaces. J. Geom. Mech. 3(4), 389–438 (2011)
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)
Bauer, M., Bruveris, M., Marsland, S., Michor, P.W.: Constructing reparameterization invariant metrics on spaces of plane curves. Differ. Geom. Appl. 34, 139–165 (2014a)
Bauer, M., Bruveris, M., Michor, P.W.: Overview of the geometries of shape spaces and diffeomorphism groups. J. Math. Imag. Vis. 50(1–2), 60–97 (2014b)
Bauer, M., Bruveris, M., Harms, P., Møller-Andersen, J.: A numerical framework for Sobolev metrics on the space of curves. SIAM J. Imag. Sci. 10(1), 47–73 (2017)
Bauer, M., Bruveris, M., Charon, N., Møller-Andersen, J.: A relaxed approach for curve matching with elastic metrics. ESAIM: Control Optim. Calc. Var. 25, 72 (2019a)
Bauer, M., Charon, N., Harms, P.: Inexact elastic shape matching in the square root normal field framework. In: Geometric Science of Information, pp. 13–20. Springer, Cham (2019b)
Bauer, M., Harms, P., Michor, P.W.: Fractional sobolev metrics on spaces of immersions. Calc. Var. Partial Differ. Equ. 59(2), 1–27 (2020a)
Bauer, M., Harms, P., Preston, S.C.: Vanishing distance phenomena and the geometric approach to sqg. Arch. Ration. Mech. Anal. 235(3), 1445–1466 (2020b)
Bauer, M., Maor, C., Michor, P.W.: Sobolev metrics on spaces of manifold valued curves. arXiv preprint arXiv:2007.13315 (2020c)
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, 139–157 (2005)
Bernal, J., Dogan, G., Hagwood, C.R.: Fast dynamic programming for elastic registration of curves. In: 2016 IEEE Conference on Computer Vision and Pattern Recognition Workshops (CVPRW), pp. 1066–1073 (2016)
Binz, E., Fischer, H.R.: The manifold of embeddings of a closed manifold. In: Differential Geometric Methods in Mathematical Physics, pp. 310–325. Springer, Berlin/Heidelberg/New York (1981)
Bruveris, M.: Completeness properties of Sobolev metrics on the space of curves. J. Geom. Mech. 7(2), 125–150 (2015)
Bruveris, M.: Optimal reparametrizations in the square root velocity framework. SIAM J. Math. Anal. 48(6), 4335–4354 (2016)
Bruveris, M., Møller-Andersen, J.: Completeness of length-weighted Sobolev metrics on the space of curves (2017). arXiv:1705.07976
Bruveris, M., Michor, P.W., Mumford, D.: Geodesic completeness for Sobolev metrics on the space of immersed plane curves. In: Forum of Mathematics, Sigma, vol. 2. Cambridge University Press, Cambridge (2014)
Celledoni, E., Eidnes, S., Schmeding, A.: Shape analysis on homogeneous spaces: a generalised srvt framework. In: The Abel Symposium, pp. 187–220. Springer (2016a)
Celledoni, E., Eslitzbichler, M., Schmeding, A.: Shape analysis on lie groups with applications in computer animation. J. Geom. Mech. 8(3), 273–304 (2016b)
Cervera, V., Mascaro, F., Michor, P.W.: The action of the diffeomorphism group on the space of immersions. Differ. Geom. Appl. 1(4), 391–401 (1991)
Charon, N., Trouvé, A.: The varifold representation of non-oriented shapes for diffeomorphic registration. SIAM J. Imag. Sci. 6(4), 2547–2580 (2013)
Charon, N., Charlier, B., Glaunès, J., Gori, P., Roussillon, P.: Fidelity metrics between curves and surfaces: currents, varifolds, and normal cycles. In: Riemannian Geometric Statistics in Medical Image Analysis, pp. 441–477. Academic Press, San Diego (2020)
Dryden, I.L., Mardia, K.V.: Statistical Shape Analysis, with Applications in R, 2nd edn. Wiley, Chichester (2016)
Durrleman, S., Fillard, P., Pennec, X., Trouvé, A., Ayache, N.: Registration, atlas estimation and variability analysis of white matter fiber bundles modeled as currents. NeuroImage 55(3), 1073–1090 (2010)
Eliashberg, Y., Polterovich, L.: Bi-invariant metrics on the group of Hamiltonian diffeomorphisms. Int. J. Math. 4(5), 727–738 (1993)
Glaunès, J., Qiu, A., Miller, M., Younes, L.: Large deformation diffeomorphic metric curve mapping. Int. J. Comput. Vis. 80(3), 317–336 (2008)
Grenander, U.: General Pattern Theory: A Mathematical Study of Regular Structures. Clarendon Press Oxford, Oxford/Clarendon/New York (1993)
Hamilton, R.S.: The inverse function theorem of Nash and Moser. Am. Math. Soc. 7(1), 65–122 (1982)
Huang, W., Gallivan, K.A., Srivastava, A., Absil, P.-A.: Riemannian optimization for registration of curves in elastic shape analysis. J. Math. Imag. Vis. 54(3), 320–343 (2016)
Huang, W., Gallivan, K.A., Srivastava, A., Absil, P.-A., et al.: Riemannian optimization for elastic shape analysis. In: Mathematical Theory of Networks and Systems. Springer (2014)
Jermyn, I.H., Kurtek, S., Laga, H., Srivastava, A.: Elastic shape analysis of three-dimensional objects. Synth. Lect. Comput. Vis. 12(1), 1–185 (2017)
Jerrard, R.L., Maor, C.: Vanishing geodesic distance for right-invariant sobolev metrics on diffeomorphism groups. Ann. Glob. Anal. Geom. 55(4), 631–656 (2019)
Kaltenmark, I., Charlier, B., Charon, N.: A general framework for curve and surface comparison and registration with oriented varifolds. In: Computer Vision and Pattern Recognition (CVPR) (2017)
Kendall, D.G.: Shape manifolds, procrustean metrics, and complex projective spaces. Bull. Lond. Math. Soc. 16(2), 81–121 (1984)
Kilian, M., Mitra, N.J., Pottmann, H.: Geometric modeling in shape space. In: ACM Transactions on Graphics (TOG), vol. 26, p. 64. ACM (2007)
Klassen, E., Srivastava, A., Mio, M., Joshi, S.H.: Analysis of planar shapes using geodesic paths on shape spaces. IEEE Trans. Pattern Anal. Mach. Intell. 26(3), 372–383 (2004)
Kurtek, S., Klassen, E., Ding, Z., Jacobson, S.W., Jacobson, J.L., Avison, M.J., Srivastava, A.: Parameterization-invariant shape comparisons of anatomical surfaces. IEEE Trans. Med. Imag. 30(3), 849–858 (2011)
Lahiri, S., Robinson, D., Klassen, E.: Precise matching of PL curves in \(\mathbb {R}^N\) in the square root velocity framework. Geom. Imag. Comput. 2(3), 133–186 (2015)
Le Brigant, A.: Computing distances and geodesics between manifold-valued curves in the SRV framework. J. Geom. Mech. 9(2), 131–156 (2017)
Le Brigant, A.: A discrete framework to find the optimal matching between manifold-valued curves. J. Math. Imag. Vis. 61(1), 40–70 (2019)
Mennucci, A.C., Yezzi, A., Sundaramoorthi, G.: Properties of Sobolev-type metrics in the space of curves. Interfaces Free Bound. 10(4), 423–445 (2008)
Michor, P.W.: Manifolds of Differentiable Mappings, vol. 3. Birkhauser and Springer (1980)
Michor, P.W.: Topics in Differential Geometry, vol. 93. American Mathematical Society, Providence (2008)
Michor, P.W., Mumford, D.: Vanishing geodesic distance on spaces of submanifolds and diffeomorphisms. Doc. Math. 10, 217–245 (2005)
Michor, P.W., Mumford, D.: Riemannian geometries on spaces of plane curves. J. Eur. Math. Soc. 8, 1–48 (2006)
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)
Mio, W., Srivastava, A., Joshi, S.: On shape of plane elastic curves. Int. J. Comput. Vis. 73(3), 307–324 (2007)
Nardi, G., Peyré, G., Vialard, F.-X.: Geodesics on shape spaces with bounded variation and Sobolev metrics. SIAM J. Imag. Sci. 9(1), 238–274 (2016)
Needham, T., Kurtek, S.: Simplifying transforms for general elastic metrics on the space of plane curves. SIAM J. Imag. Sci. 13(1), 445–473 (2020)
Roussillon, P., Glaunès, J.: Kernel metrics on normal cycles and application to curve matching. SIAM J. Imag. Sci. 9(4), 1991–2038 (2016)
Srivastava, A., Klassen, E.: Functional and Shape Data Analysis. Springer Series in Statistics. Springer, New York (2016)
Srivastava, A., Klassen, E., Joshi, S.H., Jermyn, I.H.: Shape analysis of elastic curves in Euclidean spaces. IEEE T. Pattern Anal. 33(7), 1415–1428 (2011)
Su, J., Kurtek, S., Klassen, E., Srivastava, A.: Statistical analysis of trajectories on Riemannian manifolds: bird migration, hurricane tracking and video surveillance. Ann. Appl. Stat. 8(1), 530–552 (2014)
Su, Z., Klassen, E., Bauer, M.: The square root velocity framework for curves in a homogeneous space. In: Proceedings of 2017 IEEE Conference on Computer Vision and Pattern Recognition Workshops, pp. 680–689 (2017)
Su, Z., Klassen, E., Bauer, M.: Comparing curves in homogeneous spaces. Differ. Geom. Appl. 60, 9–32 (2018)
Su, Z., Bauer, M., Preston, S.C., Laga, H., Klassen, E.: Shape analysis of surfaces using general elastic metrics. J. Math. Imag. Vis. 62, 1087–1106 (2020)
Sukurdeep, Y., Bauer, M., Charon, N.: An inexact matching approach for the comparison of plane curves with general elastic metrics. In: 2019 53rd Asilomar Conference on Signals, Systems, and Computers, pp. 512–516. IEEE (2019)
Sundaramoorthi, G., Yezzi, A., Mennucci, A.C.: Sobolev active contours. Int. J. Comput. Vis. 73(3), 345–366 (2007)
Trouvé, A.: Diffeomorphisms groups and pattern matching in image analysis. Int. J. Comput. Vis. 28(3), 213–221 (1998)
Trouvé, A., Younes, L.: Diffeomorphic matching problems in one dimension: Designing and minimizing matching functionals. In: European Conference on Computer Vision, pp. 573–587. Springer (2000a)
Trouvé, A., Younes, L.: On a class of diffeomorphic matching problems in one dimension. SIAM J. Control Optim. 39(4), 1112–1135 (2000b)
Tumpach, A.B., Drira, H., Daoudi, M., Srivastava, A.: Gauge invariant framework for shape analysis of surfaces. IEEE Trans. Pattern Anal. Mach. Intell. 38(1), 46–59 (2015)
Tumpach, A.B., Preston, S.C.: Quotient elastic metrics on the manifold of arc-length parameterized plane curves. J. Geom. Mech. 9(2), 227–256 (2017)
Younes, L.: Computable elastic distances between shapes. SIAM J. Appl. Math. 58(2), 565–586 (1998)
Younes, L.: Hybrid Riemannian metrics for diffeomorphic shape registration. Ann. Math. Sci. Appl. 3(1), 189–210 (2018)
Younes, L.: Shapes and Diffeomorphisms. Springer (2019)
Younes, L., Michor, P.W., Shah, J., Mumford, D.: A metric on shape space with explicit geodesics. Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. 19(1), 25–57 (2008)
Zhang, Z., Su, J., Klassen, E., Le, H., Srivastava, A.: Video-based action recognition using rate-invariant analysis of covariance trajectories. arXiv preprint arXiv:1503.06699 (2015)
Zhang, Z., Klassen, E., Srivastava, A.: Phase-amplitude separation and modeling of spherical trajectories. J. Comput. Graph. Stat. 27(1), 85–97 (2018a)
Zhang, Z., Su, J., Klassen, E., Le, H., Srivastava, A.: Rate-invariant analysis of covariance trajectories. J. Math. Imag. Vis. 60(8), 1306–1323 (2018b)
Acknowledgements
M. Bauer was partially supported by NSF-grant 1912037 (collaborative research in connection with NSF-grant 1912030) and NSF-grant 1953244 (collaborative research in connection with NSF-grant 1953267). N. Charon was partially supported by NSF-grant 1945224 and NSF-grant 1953267 (collaborative research in connection with NSF-grant 1953244). Eric Klassen gratefully acknowledges the support of the Simons Foundation-grant 317865.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2023 Springer Nature Switzerland AG
About this entry
Cite this entry
Bauer, M., Charon, N., Klassen, E., Le Brigant, A. (2023). Intrinsic Riemannian Metrics on Spaces of Curves: Theory and Computation. In: Chen, K., Schönlieb, CB., Tai, XC., Younes, L. (eds) Handbook of Mathematical Models and Algorithms in Computer Vision and Imaging. Springer, Cham. https://doi.org/10.1007/978-3-030-98661-2_87
Download citation
DOI: https://doi.org/10.1007/978-3-030-98661-2_87
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-98660-5
Online ISBN: 978-3-030-98661-2
eBook Packages: Mathematics and StatisticsReference Module Computer Science and Engineering