Abstract
Tensors are nowadays a common source of geometric information. In this paper, we propose to endow the tensor space with an affine-invariant Riemannian metric. We demonstrate that it leads to strong theoretical properties: the cone of positive definite symmetric matrices is replaced by a regular and complete manifold without boundaries (null eigenvalues are at the infinity), the geodesic between two tensors and the mean of a set of tensors are uniquely defined, etc.
We have previously shown that the Riemannian metric provides a powerful framework for generalizing statistics to manifolds. In this paper, we show that it is also possible to generalize to tensor fields many important geometric data processing algorithms such as interpolation, filtering, diffusion and restoration of missing data. For instance, most interpolation and Gaussian filtering schemes can be tackled efficiently through a weighted mean computation. Linear and anisotropic diffusion schemes can be adapted to our Riemannian framework, through partial differential evolution equations, provided that the metric of the tensor space is taken into account. For that purpose, we provide intrinsic numerical schemes to compute the gradient and Laplace-Beltrami operators. Finally, to enforce the fidelity to the data (either sparsely distributed tensors or complete tensors fields) we propose least-squares criteria based on our invariant Riemannian distance which are particularly simple and efficient to solve.
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References
Aubert, G. and Kornprobst, P. 2001. Mathematical Problems in Image Processing, vol. 147 of Applied Mathematical Sciences, Springer.
Basser, P., Mattiello, J., and Bihan, D.L. 1994. MR diffusion tensor spectroscopy and imaging. Biophysical Journal, 66:259–267.
Batchelor, P., Hill, D., Calamante, F., and Atkinson, D. 2001. Study of the connectivity in the brain using the full diffusion tensor from MRI. In Proc. of the 17th Int. Conf. on Information Processing in Medical Imaging (IPMI 2001), M. Insana and R. Leahy (Eds.), vol. 2082 of LNCS, Springer Verlag, pp. 121–133.
Bhatia, R. 2003. On the exponential metric increasing property. Linear Algebra and its Applications, 375:211–220.
Cazals, F. and Boissonnat, J.-D. 2001. Natural coordinates of points on a surface. Comp. Geometry Theory and Applications, 19:155–173.
Chefd'hotel, C., Tschumperlé, D., Deriche, R., and Faugeras, O. 2002. Constrained flows of matrix-valued functions: Application to diffusion tensor regularization. In Proc. of ECCV 2002, Hayden et al., (Eds.), vol. 2350 of LNCS, Springer Verlag, pp. 251–265.
Chefd'hotel, C., Tschumperlé, D., Deriche, R. and Faugeras, O. 2004. Regularizing flows for constrained matrix-valued images. J. Math. Imaging and Vision, 20(1/2):147–162.
Coulon, O., Alexander, D., and Arridge, S. 2001. A regularization scheme for diffusion tensor magnetic resonance images. In Proc. of the 17th Int. Conf. on Information Processing in Medical Imaging (IPMI 2001), M. Insana, and R. Leahy (Eds.), vol. 2082 of LNCS, Springer Verlag. pp. 92–105.
Coulon, O., Alexander, D. and Arridge, S. 2004. Diffusion tensor magnetic resonance image regularization. Medical Image Analysis, 8(1):47–67.
Fillard, P., Gilmore, J., Piven, J., Lin, W., and Gerig, G. 2003. Quantitative analysis of white matter fiber properties along geodesic paths. In Proc. of MICCAI'03, Part II, R.E. Ellis and T.M. Peters (Eds.), vol. 2879 of LNCS, Montreal, Springer Verlag, pp. 16–23.
Fletcher, P.T. and Joshi, S.C. 2004. Principal geodesic analysis on symmetric spaces: Statistics of diffusion tensors. In Computer Vision and Mathematical Methods in Medical and Biomedical Image Analysis, ECCV 2004 Workshops CVAMIA and MMBIA, Prague, Czech Republic, May 15, 2004, vol. 3117 of LNCS, pp. 87–98. Springer.
Förstner, W. and Moonen, B. 1999. A metric for covariance matrices. In Qua vadis geodesia⋯? Festschrift for Erik W. Grafarend on the occasion of his 60th Birthday, F. Krumm and V.S. Schwarze (Eds.) number 1999.6 in Tech. Report of the Dpt of Geodesy and Geoinformatics, Stuttgart University, pp. 113–128.
Gallot, S., Hulin, D., and Lafontaine, J. 1993. Riemannian Geometry, 2nd edition, Springer Verlag.
Gamkrelidze, R. (Ed.) 1991. Geometry I, vol. 28 of Encyclopaedia of Mathematical Sciences, Springer Verlag.
Gerig, G., Kikinis, R., Kübler, O., and Jolesz, F. 1992. Nonlinear anisotropic filtering of MRI data. IEEE Transactions on Medical Imaging, 11(2):221–232.
Helgason, S. 1978. Differential Geometry, Lie Groups, and Symmetric Spaces. Academic Press.
Kendall, M. and Moran, P. 1963. Geometrical Probability. No. 10 in Griffin's statistical monographs and courses. Charles Griffin & Co. Ltd.
Kendall, W. 1990. Probability, convexity, and harmonic maps with small image I: Uniqueness and fine existence. Proc. London Math. Soc., 61(2):371–406.
Kobayashi, S. and Nomizu, K. 1969. Foundations of Differential Geometry, vol. II of Interscience Tracts in Pure and Applied Mathematics. John Wiley & Sons.
Le Bihan, D., Manguin, J.-F., Poupon, C., Clark, C., Pappata, S., Molko, N., and Chabriat, H. 2001. Diffusion tensor imaging: Concepts and applications. Journal Magnetic Resonance Imaging, 13(4):534–546.
Lenglet, C., Rousson, M., Deriche, R., and Faugeras, O. 2004a. Statistics on multivariate normal distributions: A geometric approach and its application to diffusion tensor MRI. Research Report 5242, INRIA.
Lenglet, C., Rousson, M., Deriche, R., and Faugeras, O. 2004b. Toward segmentation of 3D probability density fields by surface evolution: Application to diffusion MRI. Research Report 5243, INRIA.
Meijering, E. 2002. A chronology of interpolation: From ancient astronomy to modern signal and image processing. Proceedings of the IEEE, 90(3):319–342.
Nomizu, K. 1954. Invariant affine connections on homogeneous spaces. American J. of Math., 76:33–65.
Pennec, X. 1996. L'incertitude dans les problèmes de reconnaissance et de recalage—Applications en imagerie médicale et biologie moléculaire. Thèse de sciences (PhD thesis), Ecole Polytechnique, Palaiseau (France).
Pennec, X. 1999. Probabilities and statistics on Riemannian manifolds: Basic tools for geometric measurements. In Proc. of Nonlinear Signal and Image Processing (NSIP'99), A. Cetin, L. Akarun, A. Ertuzun, M. Gurcan, and Y. Yardimci (Eds.) June 20–23, Antalya, Turkey. vol. 1, pp. 194–198. IEEE-EURASIP,
Pennec, X. 2004. Probabilities and statistics on Riemannian manifolds: A geometric approach. Research Report 5093, INRIA. Int. Journal of Mathematical Imaging and Vision (submitted).
Pennec, X. and Ayache, N. 1998. Uniform distribution, distance and expectation problems for geometric features processing. Journal of Mathematical Imaging and Vision, 9(1):49–67.
Pennec, X. and Thirion, J.-P. 1997. A framework for uncertainty and validation of 3D registration methods based on points and frames. Int. Journal of Computer Vision, 25(3):203–229.
Perona, P. and Malik, J. 1990. Scale-space and edge detection using anisotropic diffusion. IEEE Trans. Pattern Analysis and Machine Intelligence (PAMI), 12(7):629–639.
Poincaré, H. 1912. Calcul des probabilités, 2nd edition, Paris.
Rey, D., Subsol, G., Delingette, H., and Ayache, N. 2002. Automatic detection and segmentation of evolving processes in 3D medical images: Application to multiple sclerosis. Medical Image Analysis, 6(2):163–179.
Sapiro, G. 2001. Geometric Partial Differential Equations and Image Analysis. Cambridge University Press.
Sibson, R. 1981. A brief description of natural neighbour interpolation. In Interpreting Multivariate Data, V. Barnet (Ed.), John Wiley & Sons, Chichester, pp. 21–36.
Skovgaard, L. 1984. A Riemannian geometry of the multivariate normal model. Scand. J. Statistics, 11:211–223.
Thévenaz, P., Blu, T., and Unser, M. 2000. Interpolation revisited. IEEE Transactions on Medical Imaging, 19(7):739–758.
Tschumperlé, D. 2002. PDE-Based Regularization of Multivalued Images and Applications. PhD thesis, University of Nice-Sophia Antipolis.
Tschumperlé, D. and Deriche, R. 2002. Orthonormal vector sets regularization with PDE's and applications. International Journal on Computer Vision, 50(3):237–252.
Weickert, J. 1998. Anisotropic Diffusion in Image Processing. Teubner-Verlag.
Weickert, J. and Brox, T. 2002. Diffusion and regularization of vector- and matrix-valued images. In Inverse Problems, Image Analysis, and Medical Imaging., M. Nashed and O. Scherzer (Eds.), vol. 313 of Contemporary Mathematics, Providence. AMS. pp. 251–268.
Westin, C., Maier, S., Mamata, H., Nabavi, A., Jolesz, F., and Kikinis, R. 2002. Processing and visualization for diffusion tensor MRI. Medical Image Analysis, 6(2):93–108.
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Pennec, X., Fillard, P. & Ayache, N. A Riemannian Framework for Tensor Computing. Int J Comput Vision 66, 41–66 (2006). https://doi.org/10.1007/s11263-005-3222-z
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DOI: https://doi.org/10.1007/s11263-005-3222-z