Abstract
Shape analysis is ubiquitous in problems of pattern and object recognition and has developed considerably in the last decade. The use of shapes is natural in applications where one wants to compare curves independently of their parametrisation. One computationally efficient approach to shape analysis is based on the Square Root Velocity Transform (SRVT). In this paper we propose a generalised SRVT framework for shapes on homogeneous manifolds. The method opens up for a variety of possibilities based on different choices of Lie group action and giving rise to different Riemannian metrics.
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- 1.
In this paper we assume all Lie groups and Lie algebras to be finite dimensional. Note however, that many of our techniques carry over to Lie groups modelled on Hilbert spaces, [9].
- 2.
Every homogeneous space G∕H is a principal H-bundle, whence there are smooth horizontal lifts of smooth curves (depending on some choice of connection, cf. e.g. [23, Chapter 5.1]).
- 3.
In the setting of manifolds on Fréchet spaces (with which we deal here) our setting of calculus is equivalent to the so called convenient calculus (see [15]). Convenient calculus defines a map f to be smooth if it “maps smooth curves to smooth curves”, i.e. f ∘ c is smooth for any smooth curve c. This yields a calculus on infinite-dimensional spaces where smoothness does not necessarily imply continuity (though this does not happen on Fréchet spaces), we refer to [15] for a detailed exposition. Note that both calculi can handle smooth maps on intervals [a, b], see e.g. [13, 1.1] and [15, Chapter 24].
- 4.
As \(\mathrm {Imm} (I,\mathcal {M}) \subseteq C^\infty (I,\mathcal {M})\) is open and the evaluation map \(\text{ev}_{0} \colon \mathrm {Imm} (I,\mathcal {M}) \rightarrow \mathcal {M}\) is a submersion, \(\mathcal {P}_{c_0} = \text{ev}_0^{-1} (c_0)\) is a closed submanifold of \(\mathrm {Imm} (I,\mathcal {M})\) (cf. [12]).
- 5.
A submanifold N of a (possibly infinite-dimensional) manifold M is called split if it is modeled on a closed subvectorspace F of the model space E of M, such that F is complemented, i.e. E = F ⊕ G as topological vector spaces (see [12, Section 1]).
- 6.
See [12] for more information on immersions between infinite-dimensional manifolds.
- 7.
Note that there might be different reductive structures on a homogeneous manifold. We refer to [20, Section 5.1] for examples and further references.
- 8.
An alternative representation of G p,n is given by considering symmetric matrices P, n × n, with rank(P) = p and P 2 = P, [14].
References
Absil, P.-A., Mahony, R., Sepulchre, R.: Optimization Algorithms on Matrix Manifolds. Princeton University Press, Princeton (2007)
Amiri, H., Schmeding, A.: A Differentiable Monoid of Smooth Maps on Lie Groupoids (2017). arXiv:1706.04816v1
Bastiani, A.: Applications différentiables et variétés différentiables de dimension infinie. J. Anal. Math. 13, 1–114 (1964)
Bauer, M., Bruveris, M., Harms, P., Michor, P.W.: Vanishing geodesic distance for the Riemannian metric with geodesic equation the KdV-equation. Ann. Global Anal. Geom. 41(4), 461–472 (2012)
Bauer, M., Bruveris, M., Michor, P.W.: Overview of the geometries of shape spaces and diffeomorphism groups. J. Math. Imaging Vis. 50(1), 1–38 (2014)
Bauer, M., Bruveris, M., Marsland, S., Michor, P.W.: Constructing reparameterization invariant metrics on spaces of plane curves. Differ. Geom. Appl. 34, 139–165 (2014)
Bauer, M., Eslitzbichler, M., Grasmair, M.: Landmark-guided elastic shape analysis of human character motions. Inverse Prob. Imaging 11(4), 601–621 (2015). https://doi.org/10.3934/ipi.2017028
Bruveris, M.: Optimal reparametrizations in the square root velocity framework. SIAM J. Math. Anal. 48(6), 4335–4354 (2016)
Celledoni, E., Eslitzbichler, M., Schmeding, A.: Shape analysis on Lie groups with applications in computer animation. J. Geom. Mech. 8(3), 273–304 (2016)
Celledoni, E., Owren, B.: On the implementation of Lie group methods on the Stiefel manifold. Numer. Algorithm. 32(2–4), 163–183 (2003)
Gallot, S., Hulin, D., Lafontaine, J.: Riemannian Geometry. Universitext, 3rd edn. Springer, Berlin (2004)
Glöckner, H.: Fundamentals of Submersions and Immersions Between Infinite-Dimensional Manifolds (2015). arXiv:1502.05795v3 [math]
Glöckner, H.: Regularity Properties of Infinite-Dimensional Lie Groups, and Semiregularity (2015). arXiv:1208.0715v3
Huper, K., Leite, F.: On the geometry of rolling and interpolation curves on S n, SOn, and Grassmann manifolds. J. Dyn. Control. Syst. 13, 467–502 (2007)
Kriegl, A., Michor, P.W.: The convenient setting of global analysis. In: Mathematical Surveys and Monographs, vol. 53. American Mathematical Society, Providence
Kobayashi, S., Nomizu, K. Foundations of Differential Geometry, vol. II. Interscience Tracts in Pure and Applied Mathematics, no. 15, vol. II. Interscience Publishers John Wiley, New York/London/Sydney (1969)
Knapp, A.W.: Lie groups beyond an introduction. In: Progress in Mathematics, vol. 140, 2nd edn. Birkhäuser, Boston (2002)
Le Brigant, A.: Computing distances and geodesics between manifold-valued curves in the SRV framework. J. Geom. Mech. 9(2) (2017). https://doi.org/10.3934/jgm.2017005
Michor, P.W.: Manifolds of Differentiable Mappings. In: Shiva Mathematics Series, vol. 3. Shiva Publishing Ltd., Nantwich (1980)
Munthe-Kaas, H., Verdier, O.: Integrators on homogeneous spaces: isotropy choice and connections. Found. Comput. Math. 16(4), 899–939 (2016)
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. (JEMS) 8(1), 1–48 (2006)
Ortega, J.-P., Ratiu, T.S.: Momentum maps and Hamiltonian reduction. In: Progress in Mathematics, vol. 222. Birkhäuser Boston, Inc., Boston (2004)
Sharpe, R.W.: Differential geometry. In: Graduate Texts in Mathematics, vol. 166. Springer, New York (1997). Cartan’s generalization of Klein’s Erlangen program, With a foreword by S. S. Chern
Su, Z., Klassen, E., Bauer, M.: Comparing Curves in Homogeneous Spaces (2017). 1712.04586v1
Su, Z., Klassen, E., Bauer, M.: The square root velocity framework for curves in a homogeneous space. In: 2017 IEEE Conference on Computer Vision and Pattern Recognition Workshops (CVPRW), pp. 680–689. IEEE (2017)
Srivastava, A., Klassen, E., Joshi, S., Jermyn, I.: Shape analysis of elastic curves in euclidean spaces. IEEE Trans. Pattern Anal. Mach. Intell. 33, 1415–1428 (2011)
Sebastian, T.B., Klein, P.N., Kimia, B.B.: On aligning curves. IEEE Trans. Pattern Anal. Mach. Intell. 25(1), 116–125 (2003)
Su, J., Kurtek, S., Klassen, E., Srivastava, A.: Statistical analysis of trajectories on Riemmannian manifolds: bird migration, hurricane tracking and video surveillance. Ann. Appl. Stat. 8(2), 530–552 (2014)
Acknowledgements
This work was supported by the Norwegian Research Council, and by the European Union’s Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie, grant agreement No. 691070.
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Celledoni, E., Eidnes, S., Schmeding, A. (2018). Shape Analysis on Homogeneous Spaces: A Generalised SRVT Framework. In: Celledoni, E., Di Nunno, G., Ebrahimi-Fard, K., Munthe-Kaas, H. (eds) Computation and Combinatorics in Dynamics, Stochastics and Control. Abelsymposium 2016. Abel Symposia, vol 13. Springer, Cham. https://doi.org/10.1007/978-3-030-01593-0_7
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