, PhD of Vincent Arsigny [Ars06] and in the research report [APA06]. In this preliminary work, the 'group geodesics' were simply defined as left translations of one-parameters subgroups without further justification. [PA12] extended this work by reformulating and rigorously justifying 'group geodesics' as the geodesics of the canonical Cartan-Schouten connections, The barycentric definition of bi-invariant means on Lie groups based on one-parameter subgroups was developed during the

D. Alekseevsky and A. Arvanitoyeorgos, Riemannian flag manifolds with homogeneous geodesics, Transactions of the American Mathematical Society, vol.359, issue.8, pp.3769-3789, 2007.

V. Arsigny, O. Commowick, N. Ayache, and X. Pennec, A fast and log-Euclidean polyaffine framework for locally linear registration, J. of Math. Imaging and Vision (JMIV), vol.33, issue.2, pp.222-238, 2009.
URL : https://hal.archives-ouvertes.fr/inria-00616084

V. Arsigny, O. Commowick, X. Pennec, and N. Ayache, A log-Euclidean framework for statistics on diffeomorphisms, Proc. of the 9th International Conference on Medical Image Computing and Computer Assisted Intervention (MICCAI'06), Part I, number 4190 in LNCS, vol.2, pp.924-931, 2006.
URL : https://hal.archives-ouvertes.fr/inria-00615594

J. Ashburner and . Friston, Voxel-based morphometry -the methods, Neuroimage, vol.11, issue.6, pp.805-821, 2000.

M. Araudon and X. Li, Barycenters of measures transported by stochastic flows. The annals of probability, vol.33, pp.1509-1543, 2005.

V. Arsigny, X. Pennec, and N. Ayache, Bi-invariant means in lie groups. application to left-invariant polyaffine transformations, 2006.
URL : https://hal.archives-ouvertes.fr/inria-00071383

, Vladimir Igorevich Arnol'd. Mathematical methods of classical mechanics, 1979.

M. Arnaudon, Espérances conditionnelles et C-martingales dans les variétés, vol.1583, pp.300-311, 1994.

M. Arnaudon, Barycentres convexes et approximations des martingales continues dans les variétés, vol.1613, pp.70-85, 1995.

V. Arsigny, Processing Data in Lie Groups: An Algebraic Approach. Application to Non-Linear Registration and Diffusion Tensor MRI, École polytechnique, 2006.
URL : https://hal.archives-ouvertes.fr/tel-00121162

J. Ashburner, A fast diffeomorphic image registration algorithm, NeuroImage, vol.38, issue.1, pp.95-113, 2007.

M. Bauer, M. Bruveris, and P. W. Michor, Uniqueness of the Fisher-Rao metric on the space of smooth densities, 2014.

M. Bossa, M. Hernandez, and S. Olmos, Contributions to 3D diffeomorphic atlas estimation: Application to brain images, Proc. of Medical Image Computing and Computer-Assisted Intervention, vol.4792, pp.667-674, 2007.

P. Buser and H. Karcher, Gromov's almost flat manifolds. Number 81 in Astérisque. Société mathématique de France, 1981.

. Mirza-faisal, . Beg, I. M. Michael, A. Miller, L. Trouvé et al., Computing large deformation metric mappings via geodesic flows of diffeomorphisms, Int. Journal of Computer Vision, vol.61, issue.2, pp.139-157, 2005.

M. Bossa, E. Zacur, and S. Olmos, On changing coordinate systems for longitudinal tensor-based morphometry, Proc. of Spatio Temporal Image Analysis Workshop, p.44, 2010.

P. Cachier and N. Ayache, Isotropic energies, filters and splines for vector field regularization, Journal of Mathematical Imaging and Vision, vol.20, issue.3, pp.251-265, 2004.

E. Cachier, D. Bardinet, X. Dormont, N. Pennec, and . Ayache, Iconic feature based nonrigid registration: The pasha algorithm. Computer Vision and Image Understanding, vol.89, pp.272-298, 2003.
URL : https://hal.archives-ouvertes.fr/inria-00615633

D. M. Cash, C. Frost, L. O. Iheme, M. Devrimünay, J. Kandemir et al., Assessing atrophy measurement techniques in dementia: Results from the MIRIAD atrophy challenge, NeuroImage, vol.123, pp.149-164, 2015.
URL : https://hal.archives-ouvertes.fr/hal-01203573

J. Manuel-corcuera and W. S. Kendall, Riemannian barycentres and geodesic convexity, Math. Prob. Camb. Phil. Soc, vol.127, pp.253-269, 1999.

E. Cartan and J. A. Schouten, On the geometry of the groupmanifold of simple and semi-simple groups, Proc. Akad. Wekensch, vol.29, pp.803-815, 1926.

C. Manfredo-do, Riemannian Geometry. Mathematics. Birkhäuser, 1992.

M. Emery and G. Mokobodzki, Sur le barycentre d'une probabilité dans une variété, Lect. Notes in Math, vol.1485, pp.220-233, 1991.

A. Peter, N. Freeborough, M. Paul, and . Thompson, The boundary shift integral: an accurate and robust measure of cerebral volume changes from registered repeat MRI, Philip Scheltens, and, vol.16, p.67, 1997.

. +-16]-sebastiano, M. Ferraris, P. Lorenzi, M. Daga, T. Modat et al., Accurate small deformation exponential approximant to integrate large velocity fields: Application to image registration, Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition Workshops, pp.17-24, 2016.

J. Gallier, Logarithms and Square Roots of Real Matrices, 2008.

A. Vladimirovich and G. , Algebraic properties of covariant derivative and composition of exponential maps, Matematicheskie Trudy, vol.9, issue.1, pp.3-20, 2006.

S. Gallot, D. Hulin, and J. Lafontaine, Riemannian Geometry, 1993.
URL : https://hal.archives-ouvertes.fr/hal-00002870

R. Godement, Introductionà la Théorie des Groupes de Lie, Tomes I et II. Publications Mathématiques de l'Université, 1982.

D. Groisser, Newton's method, zeroes of vector fields, and the Riemannian center of mass, Adv. in Applied Math, vol.33, pp.95-135, 2004.

M. Hernandez, M. Bossa, and S. Olmos, Registration of anatomical images using paths of diffeomorphisms parameterized with stationary vector field flows, International Journal of Computer Vision, vol.85, pp.291-306, 2009.

. Sheung-hun, N. J. Cheng, C. S. Higham, A. J. Kenney, and . Laub, Approximating the logarithm of a matrix to specified accuracy, SIAM J. Matrix Anal. Appl, vol.22, issue.4, pp.1112-1125, 2001.

, Sigurdur Helgason. Differential Geometry, Lie groups, and Symmetric Spaces, 1978.

N. J. Higham, The scaling and squaring method for the matrix exponential revisited, SIAM Journal on Matrix Analysis and Applications, vol.26, issue.4, pp.1179-1193, 2005.

E. Hairer, C. Lubich, and G. Wanner, Geometric numerical integration : structure preserving algorithm for ordinary differential equations, volume 31 of Springer series in computational mathematics, 2002.

A. Iserles, Z. Hans, . Munthe-kaas, P. Syvert, A. Norsett et al., Lie-group methods, Acta numerica, vol.9, pp.215-365, 2000.
URL : https://hal.archives-ouvertes.fr/hal-01328729

W. S. Kendall, Convexity and the hemisphere, Journal of the London Mathematical Society, vol.43, issue.2, pp.567-576, 1991.

W. S. Kendall, The propeller: a counterexample to a conjectured criterion for the existence of certain harmonic functions, Journal of the London Mathematical Society, vol.46, pp.364-374, 1992.

C. S. Kenney and A. J. Laub, Condition estimates for matrix functions, SIAM J. Matrix Anal. Appl, vol.10, pp.191-209, 1989.

W. R. Klingenberg and . Geometry, , 1982.

A. Kheyfets, A. Warner, G. Miller, and . Newton, Schild's ladder parallel transport procedure for an arbitrary connection, International Journal of Theoretical Physics, vol.39, issue.12, pp.2891-2898, 2000.

B. Kolev, Groupes de lie et mécanique, Notes of a Master course in 2006-2007 at Université de Provence, 2007.

B. A. Khesin and R. Wendt, The Geometry of Infinite Dimensional Lie groups, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge / A Series of Modern Surveys in Mathematics, vol.51, 2009.

M. Lorenzi, N. Ayache, G. B. Frisoni, and X. Pennec, LCC-Demons: a robust and accurate symmetric diffeomorphic registration algorithm, NeuroImage, vol.81, issue.1, pp.470-483, 2013.
URL : https://hal.archives-ouvertes.fr/hal-00819895

S. L. Algebra, Graduate Texts in Mathematics, 2004.

M. Lorenzi, N. Ayache, and X. Pennec, Schild's ladder for the parallel transport of deformations in time series of images, IPMI -22nd International Conference on Information Processing in Medical Images-2011, vol.6801, pp.463-474, 2011.

M. Lorenzi, N. Ayache, and X. Pennec, Regional flux analysis for discovering and quantifying anatomical changes: An application to the brain morphometry in Alzheimer's disease, NeuroImage, vol.115, pp.224-234, 2015.

H. and T. Laquer, Invariant affine connections on Lie groups, Transactions of the, vol.331, pp.541-551, 1992.

M. Lorenzi, G. B. Frisoni, N. Ayache, and X. Pennec, Mapping the effects of A? 1?42 levels on the longitudinal changes in healthy aging: hierarchical modeling based on stationary velocity fields, Medical Image Computing and Computer-Assisted Intervention -MICCAI 2011, vol.6893, pp.663-670, 2011.

M. Lorenzi and X. Pennec, Geodesics, parallel transport & oneparameter subgroups for diffeomorphic image registration, International Journal of Computer Vision, vol.105, issue.2, pp.111-127, 2013.
URL : https://hal.archives-ouvertes.fr/hal-00813835

M. Lorenzi and X. Pennec, Efficient parallel transport of deformations in time series of images: from Schild's to pole ladder, Journal of Mathematical Imaging and Vision, vol.50, issue.1-2, pp.5-17, 2014.

M. Lorenzi, X. Pennec, G. B. Frisoni, and N. Ayache, Disentangling normal aging from Alzheimer's disease in structural MR images, Neurobiology of Aging, 2014.

A. Medina, Groupes de Lie munis de pseudo-métriques de Riemann bi-invariantes. Séminaire de géométrie différentielle, 1981.

J. Milnor, Remarks on infinite-dimensional Lie groups, Relativity, Groups and Topology, pp.1009-1057, 1984.

M. Moakher, Means and averaging in the group of rotations, SIAM Journal on Matrix Analysis and Applications, vol.24, issue.1, pp.1-16, 2002.

N. Miolane and X. Pennec, Computing bi-invariant pseudometrics on Lie groups for consistent statistics, Entropy, vol.17, issue.4, pp.1850-1881, 2015.
URL : https://hal.archives-ouvertes.fr/hal-01133922

K. Mcleod, A. Prakosa, T. Mansi, M. Sermesant, and X. Pennec, Mihaela Pop, Kawal Rhode, Maxime Sermesant, and Alistair Young, editors, Statistical Atlases and Computational Models of the Heart. Imaging and Modelling Challenges, International Journal of Computer Vision, vol.7085, issue.1, pp.92-111, 2011.

A. Medina and P. Revoy, Annales scientifiques de l'Ecole normale supérieure, vol.18, pp.553-561, 1985.

M. Modat, G. R. Ridgway, P. Daga, M. J. Cardoso, D. J. Hawkes et al., Logeuclidean free-form deformation, Proc. of SPIE Medical Imaging, 2011.

K. S. Charles-w-misner, J. A. Thorne, and . Wheeler, Gravitation, 1973.

C. Moler and C. Van-loan, Nineteen dubious ways to compute the exponential of a matrix, twenty-five years later, SIAM review, vol.45, issue.1, pp.3-49, 2003.

Y. G. Nikonorov, Evgenii Dmirtievich Rodionov, and Viktor Vladimirovich Slavskii. Geometry of homogeneous Riemannian manifolds, Journal of Mathematical Sciences, vol.146, issue.6, pp.6313-6390, 2007.

X. Pennec and V. Arsigny, Exponential barycenters of the canonical Cartan connection and invariant means on Lie groups, Matrix Information Geometry, pp.123-168, 2012.
URL : https://hal.archives-ouvertes.fr/hal-00699361

J. Peyrat, H. Delingette, M. Sermesant, and X. Pennec, Registration of 4D time-series of cardiac images with multichannel diffeomorphic demons, Medical Image Computing and Computer-Assisted Intervention -MICCAI, vol.5242, pp.972-979, 2008.
URL : https://hal.archives-ouvertes.fr/inria-00616122

X. Pennec, L'incertitude dans les problèmes de reconnaissance et de recalage -Applications en imagerie médicale et biologie moléculaire, 1996.

X. Pennec, Computing the mean of geometric features -application to the mean rotation, INRIA, 1998.
URL : https://hal.archives-ouvertes.fr/inria-00073318

X. Pennec, Intrinsic statistics on Riemannian manifolds: Basic tools for geometric measurements, Journal of Mathematical Imaging and Vision, vol.25, issue.1, pp.127-154, 2004.
URL : https://hal.archives-ouvertes.fr/inria-00614994

X. Pennec, Statistical Computing on Manifolds for Computational Anatomy. Habilitationà diriger des recherches, 2006.
URL : https://hal.archives-ouvertes.fr/tel-00633163

X. Pennec, Parallel transport with pole ladder: a third order scheme in affine connection spaces which is exact in affine symmetric spaces, 2018.
URL : https://hal.archives-ouvertes.fr/hal-01799888

P. Mikhail-mikhailovich, Geometry VI: Riemannian Geometry. Encyclopedia of mathematical science, 2001.

A. Qiu, L. Younes, I. Michael, J. G. Miller, and . Csernansky, Parallel transport in diffeomorphisms distinguishes the time-dependent pattern of hippocampal surface deformation due to healthy aging and the dementia of the alzheimer's type, NeuroImage, vol.40, issue.1, pp.68-76, 2008.

A. Rao, R. Chandrashekara, G. I. Sanchez-ortiz, R. Mohiaddin, P. Aljabar et al., Spatial transformation of motion and deformation fields using nonrigid registration, IEEE Transactions on Medical Imaging, vol.23, issue.9, pp.1065-1076, 2004.

R. William-r-riddle, M. Li, S. C. Fitzpatrick, . Donlevy, M. Benoit et al., Characterizing changes in mr images with color-coded jacobians, Magnetic resonance imaging, vol.22, issue.6, pp.769-777, 2004.

M. Stephen, N. D. Smith, M. Stefano, P. Jenkinson, and . Matthews, Normalized accurate measurement of longitudinal brain change, Journal of computer assisted tomography, vol.25, issue.3, pp.466-475, 2001.

C. Seiler, X. Pennec, and M. Reyes, Geometryaware multiscale image registration via OBBTree-based polyaffine logdemons, Medical Image Computing and Computer-Assisted Intervention -MICCAI 2011, vol.6893, pp.631-638, 2011.
URL : https://hal.archives-ouvertes.fr/inria-00616215

S. Sternberg, Lectures on Differential Geometry, 1964.

A. Trouvé, Diffeomorphisms groups and pattern matching in image analysis, International Journal of Computer Vision, vol.28, issue.3, pp.213-221, 1998.

T. Vercauteren, X. Pennec, A. Perchant, and N. Ayache, Non-parametric diffeomorphic image registration with the Demons algorithm, Medical Image Computing and Computer-Assisted Intervention -MICCAI 2007, pp.319-326, 2007.
URL : https://hal.archives-ouvertes.fr/inria-00166123

T. Vercauteren and X. Pennec, Aymeric Perchant, and Nicholas Ayache. Symmetric Log-domain diffeomorphic registration: A Demonsbased approach, Proc. of Medical Image Computing and Computer-Assisted Intervention -MICCAI, vol.5241, pp.754-761, 2008.

T. Vercauteren and X. Pennec, Aymeric Perchant, and Nicholas Ayache. Diffeomorphic demons: Efficient non-parametric image registration, NeuroImage, vol.45, issue.1, pp.61-72, 2009.

M. Wüstner, A connected Lie group equals the square of the exponential image, Journal of Lie Theory, vol.13, pp.307-309, 2003.

L. Younes, Jacobi fields in groups of diffeomorphisms and applications, Quarterly of applied mathematics, pp.113-134, 2007.

M. Zefran, V. Kumar, and C. Croke, Metrics and connections for rigid-body kinematics, Int. Journal of Robotics Research, vol.18, issue.2, pp.243-258, 1999.