Abstract
The analysis of manifold-valued data requires efficient tools from Riemannian geometry to cope with the computational complexity at stake. This complexity arises from the always-increasing dimension of the data, and the absence of closed-form expressions to basic operations such as the Riemannian logarithm. In this paper, we adapt a generic numerical scheme recently introduced for computing parallel transport along geodesics in a Riemannian manifold to finite-dimensional manifolds of diffeomorphisms. We provide a qualitative and quantitative analysis of its behavior on high-dimensional manifolds, and investigate an application with the prediction of brain structures progression.
M. Louis and A. Bône—Equal contributions.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsReferences
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)
Durrleman, S., Allassonnière, S., Joshi, S.: Sparse adaptive parameterization of variability in image ensembles. Int. J. Comput. Vis. 101(1), 161–183 (2013)
Fletcher, T.: Geodesic regression and the theory of least squares on Riemannian manifolds. Int. J. Comput. Vis. 105(2), 171–185 (2013)
Lenglet, C., Rousson, M., Deriche, R., Faugeras, O.: Statistics on the manifold of multivariate normal distributions: theory and application to diffusion tensor MRI processing. J. Math. Imaging Vis. 25(3), 423–444 (2006)
Lorenzi, M., Ayache, N., Pennec, X.: Schild’s ladder for the parallel transport of deformations in time series of images. In: Székely, G., Hahn, H.K. (eds.) IPMI 2011. LNCS, vol. 6801, pp. 463–474. Springer, Heidelberg (2011). doi:10.1007/978-3-642-22092-0_38
Lorenzi, M., Pennec, X.: Geodesics, parallel transport & one-parameter subgroups for diffeomorphic image registration. Int. J. Comput. Vis. 105(2), 111–127 (2013)
Marco, L., Pennec, X.: Parallel transport with pole ladder: application to deformations of time series of images. In: Nielsen, F., Barbaresco, F. (eds.) GSI 2013. LNCS, vol. 8085, pp. 68–75. Springer, Heidelberg (2013). doi:10.1007/978-3-642-40020-9_6
Louis, M., Charlier, B., Jusselin, P., Pal, S., Durrleman, S.: A fanning scheme for the parallel transport along geodesics on Riemannian manifolds, July 2017. https://hal.archives-ouvertes.fr/hal-01560787
Micheli, M.: The differential geometry of landmark shape manifolds: metrics, geodesics, and curvature. Ph.D. thesis, Providence, RI, USA (2008). aAI3335682
Miller, M.I., Trouvé, A., Younes, L.: Geodesic shooting for computational anatomy. J. Math. Imaging Vis. 24(2), 209–228 (2006)
Schiratti, J.B., Allassonnière, S., Colliot, O., Durrleman, S.: Learning spatiotemporal trajectories from manifold-valued longitudinal data. In: NIPS (2015)
Younes, L.: Jacobi fields in groups of diffeomorphisms and applications. Q. Appl. Math. 65(1), 113–134 (2007)
Zhang, M., Fletcher, P.: Probabilistic principal geodesic analysis. In: Advances in Neural Information Processing Systems, vol. 26, pp. 1178–1186 (2013)
Author information
Authors and Affiliations
Consortia
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2017 Springer International Publishing AG
About this paper
Cite this paper
Louis, M., Bône, A., Charlier, B., Durrleman, S., The Alzheimer’s Disease Neuroimaging Initiative. (2017). Parallel Transport in Shape Analysis: A Scalable Numerical Scheme. In: Nielsen, F., Barbaresco, F. (eds) Geometric Science of Information. GSI 2017. Lecture Notes in Computer Science(), vol 10589. Springer, Cham. https://doi.org/10.1007/978-3-319-68445-1_4
Download citation
DOI: https://doi.org/10.1007/978-3-319-68445-1_4
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-68444-4
Online ISBN: 978-3-319-68445-1
eBook Packages: Computer ScienceComputer Science (R0)