Abstract
In inter-subject registration, one often lacks a good model of the transformation variability to choose the optimal regularization. Some works attempt to model the variability in a statistical way, but the re-introduction in a registration algorithm is not easy. In this paper, we interpret the elastic energy as the distance of the Green-St Venant strain tensor to the identity, which reflects the deviation of the local deformation from a rigid transformation. By changing the Euclidean metric for a more suitable Riemannian one, we define a consistent statistical framework to quantify the amount of deformation. In particular, the mean and the covariance matrix of the strain tensor can be consistently and efficiently computed from a population of non-linear transformations. These statistics are then used as parameters in a Mahalanobis distance to measure the statistical deviation from the observed variability, giving a new regularization criterion that we called the statistical Riemannian elasticity. This new criterion is able to handle anisotropic deformations and is inverse-consistent. Preliminary results show that it can be quite easily implemented in a non-rigid registration algorithms.
Chapter PDF
Similar content being viewed by others
Keywords
- Strain Tensor
- Registration Algorithm
- Rigid Transformation
- Regularization Criterion
- Deformable Image Registration
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
References
Bajcsy, R., Kovačič, S.: Multiresolution elastic matching. Computer Vision, Graphics and Image Processing 46, 1–21 (1989)
Christensen, G.E., Joshi, S.C., Miller, M.I.: Volumetric transformation of brain anatomy. IEEE Trans. Med. Imaging 16(6), 864–877 (1997)
Thirion, J.-P.: Image matching as a diffusion process: an analogy with maxwell’s demons. Medical Image Analysis 2(3) (1998)
Modersitzki, J.: Numerical Methods for Image Registration. Numerical Mathematics and Scientific Computations. Oxford University Press, Oxford (2004)
Lester, H., Arridge, S.R., Jansons, K.M., Lemieux, L., Hajnal, J.V., Oatridge, A.: Non-linear registration with the variable viscosity fluid algorithm. In: Kuba, A., Sámal, M., Todd-Pokropek, A. (eds.) IPMI 1999. LNCS, vol. 1613, pp. 238–251. Springer, Heidelberg (1999)
Stefanescu, R., Pennec, X., Ayache, N.: Grid powered nonlinear image registration with locally adaptive regularization. Med. Image Anal. 8(3), 325–342 (2004)
Thompson, P.M., Mega, M.S., Narr, K.L., Sowell, E.R., Blanton, R.E., Toga, A.W.: Brain image analysis and atlas construction. In: Fitzpatrick, M., Sonka, M. (eds.) Handbook of Medical Image Proc. and Analysis, ch. 17. SPIE (2000)
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, 1014–1025 (2003)
Fillard, P., Arsigny, V., Pennec, X., Thompson, P., Ayache, N.: Extrapolation of sparse tensor fields: Application to the modeling of brain variability. In: Christensen, G.E., Sonka, M. (eds.) IPMI 2005. LNCS, vol. 3565, pp. 27–38. Springer, Heidelberg (2005)
Ciarlet, P.G.: Mathematical elasticity. Three-dimensionnal elasticity, vol. 1. Elsevier Science B.V, Amsterdam (1988)
Christensen, G.E., Johnson, H.: Consistent image registration. IEEE Trans. Med. Imaging 20(7), 568–582 (2001)
Batchelor, P., Moakher, M., Atkinson, D., Calamante, F., Connelly, A.: A rigorous framework for diffusion tensor calculus. Mag. Res. in Med. 53, 221–225 (2005)
Fletcher, P.T., Joshi, S.C.: Principal geodesic analysis on symmetric spaces: Statistics of diffusion tensors. In: Sonka, M., Kakadiaris, I.A., Kybic, J. (eds.) CVAMIA/MMBIA 2004. LNCS, vol. 3117, pp. 87–98. Springer, Heidelberg (2004)
Lenglet, C., Rousson, M., Deriche, R., Faugeras, O.: Statistics on multivariate normal distributions: A geometric approach and its application to diffusion tensor MRI. Research Report 5242, INRIA (2004)
Pennec, X., Fillard, P., Ayache, N.: A Riemannian framework for tensor computing. IJCV 65(1) (October 2005); Also as INRIA Research Report 5255
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)
Freed, A.D.: Natural strain. J. of Eng. Materials & Technology 117, 379–385 (1995)
Woods, R.P.: Characterizing volume and surface deformations in an atlas framework: theory, applications, and implementation. NeuroImage 18(3), 769–788 (2003)
Arsigny, V., Pennec, X., Fillard, P., Ayache, N.: Dispositif perfectionné de traitement ou de production d’images de tenseurs. Patent filing 0503483 (April 2005)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2005 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Pennec, X., Stefanescu, R., Arsigny, V., Fillard, P., Ayache, N. (2005). Riemannian Elasticity: A Statistical Regularization Framework for Non-linear Registration. In: Duncan, J.S., Gerig, G. (eds) Medical Image Computing and Computer-Assisted Intervention – MICCAI 2005. MICCAI 2005. Lecture Notes in Computer Science, vol 3750. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11566489_116
Download citation
DOI: https://doi.org/10.1007/11566489_116
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-29326-2
Online ISBN: 978-3-540-32095-1
eBook Packages: Computer ScienceComputer Science (R0)