Abstract
In this article, we focus on the computation of statistics of invertible geometrical deformations (i.e., diffeomorphisms), based on the generalization to this type of data of the notion of principal logarithm. Remarkably, this logarithm is a simple 3D vector field, and is well-defined for diffeomorphisms close enough to the identity. This allows to perform vectorial statistics on diffeomorphisms, while preserving the invertibility constraint, contrary to Euclidean statistics on displacement fields. We also present here two efficient algorithms to compute logarithms of diffeomorphisms and exponentials of vector fields, whose accuracy is studied on synthetic data. Finally, we apply these tools to compute the mean of a set of diffeomorphisms, in the context of a registration experiment between an atlas an a database of 9 T1 MR images of the human brain.
Chapter PDF
Similar content being viewed by others
References
Arsigny, V., Commowick, O., Pennec, X., Ayache, N.: A Log-Euclidean polyaffine framework for locally rigid or affine registration. In: Pluim, J.P.W., Likar, B., Gerritsen, F.A. (eds.) WBIR 2006. LNCS, vol. 4057, pp. 120–127. Springer, Heidelberg (2006)
Arsigny, V., Fillard, P., Pennec, X., Ayache, N.: Fast and simple calculus on tensors in the Log-Euclidean framework. In: Duncan, J.S., Gerig, G. (eds.) MICCAI 2005. LNCS, vol. 3749, pp. 115–122. Springer, Heidelberg (2005)
Beg, M.F., Miller, M.I., Trouvé, A., Younes, L.: Computing large deformation metric mappings via geodesic flows of diffeomorphisms. Int. Jour. Comp. Vis. 61(2), 139–157 (2005)
Camion, V., Younes, L.: Geodesic interpolating splines. In: Figueiredo, M., Zerubia, J., Jain, A.K. (eds.) EMMCVPR 2001. LNCS, vol. 2134, pp. 513–527. Springer, Heidelberg (2001)
Chefd’hotel, C., Hermosillo, G., Faugeras, O.: Flows of diffeomorphisms for multimodal image registration. In: Proc. of ISBI (2002)
Commowick, O., Stefanescu, R., Fillard, P., Arsigny, V., Ayache, N., Pennec, X., Malandain, G.: Incorporating Statistical Measures of Anatomical Variability in Atlas-to-Subject Registration for Conformal Brain Radiotherapy. In: Duncan, J.S., Gerig, G. (eds.) MICCAI 2005. LNCS, vol. 3750, pp. 927–934. Springer, Heidelberg (2005)
Higham, N.J.: The scaling and squaring method for the matrix exponential revisited. SIAM J. Matrix Anal. Appl. 26(4), 1179–1193 (2005)
Marsland, S., Twining, C.J.: Constructing diffeomorphic representations for the groupwise analysis of nonrigid registrations of medical images. IEEE Trans. Med. Imaging 23(8), 1006–1020 (2004)
Pennec, X., Stefanescu, R., Arsigny, V., Fillard, P., Ayache, N.: Riemannian Elasticity: A Statistical Regularization Framework for Non-linear Registration. In: Duncan, J.S., Gerig, G. (eds.) MICCAI 2005. LNCS, vol. 3750, pp. 943–950. Springer, Heidelberg (2005)
Rueckert, D., Frangi, A.F., Schnabel, J.A.: Automatic construction of 3D statistical deformation models of the brain using non-rigid registration. IEEE TMI 22(8), 1014–1025 (2003)
Stefanescu, R., Pennec, X., Ayache, N.: Grid powered nonlinear image registration with locally adaptive regularization. MedI. A 88(3), 325–342 (2004)
Sternberg, S.: Lectures on Differential Geometry. Prentice Hall Mathematics Series. Prentice Hall Inc., Englewood Cliffs (1964)
Tenenbaum, M., Pollard, H.: Ordinary Differential Equations. Dover (1985)
Trouvé, A.: Diffeomorphisms groups and pattern matching in image analysis. International Journal of Computer Vision 28(3), 213–221 (1998)
Vaillant, M., Miller, M.I., Younes, L., Trouvé, A.: Statistics on diffeomorphisms via tangent space representations. NeuroImage 23, S161–S169 (2004)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2006 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Arsigny, V., Commowick, O., Pennec, X., Ayache, N. (2006). A Log-Euclidean Framework for Statistics on Diffeomorphisms. In: Larsen, R., Nielsen, M., Sporring, J. (eds) Medical Image Computing and Computer-Assisted Intervention – MICCAI 2006. MICCAI 2006. Lecture Notes in Computer Science, vol 4190. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11866565_113
Download citation
DOI: https://doi.org/10.1007/11866565_113
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-44707-8
Online ISBN: 978-3-540-44708-5
eBook Packages: Computer ScienceComputer Science (R0)