caa a22 4500 46791480X CHVBK 20180406152859.0 cr unu---uuuuu 170328e20060101xx s 000 0 eng 10.1007/s11263-005-3222-z doi (NATIONALLICENCE)springer-10.1007/s11263-005-3222-z A Riemannian Framework for Tensor Computing [Elektronische Daten] [Xavier Pennec, Pierre Fillard, Nicholas Ayache] 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. Springer Science + Business Media, Inc, 2006 tensors nationallicence diffusion tensor MRI nationallicence regularization nationallicence interpolation nationallicence extrapolation nationallicence PDE nationallicence Riemannian manifold nationallicence affine-invariant metric nationallicence Pennec Xavier EPIDAURE/ASCLEPIOS Project-team, INRIA Sophia-Antipolis, 2004 Route des Lucioles BP 93, F-06902, Sophia Antipolis Cedex, France aut Fillard Pierre EPIDAURE/ASCLEPIOS Project-team, INRIA Sophia-Antipolis, 2004 Route des Lucioles BP 93, F-06902, Sophia Antipolis Cedex, France aut Ayache Nicholas EPIDAURE/ASCLEPIOS Project-team, INRIA Sophia-Antipolis, 2004 Route des Lucioles BP 93, F-06902, Sophia Antipolis Cedex, France aut International Journal of Computer Vision Kluwer Academic Publishers 66/1(2006-01-01), 41-66 0920-5691 66:1<41 2006 66 11263 https://doi.org/10.1007/s11263-005-3222-z text/html Onlinezugriff via DOI 1 research-article jats NATIONALLICENCE 856 40 https://doi.org/10.1007/s11263-005-3222-z text/html Onlinezugriff via DOI NATIONALLICENCE 700 1- Pennec Xavier EPIDAURE/ASCLEPIOS Project-team, INRIA Sophia-Antipolis, 2004 Route des Lucioles BP 93, F-06902, Sophia Antipolis Cedex, France aut NATIONALLICENCE 700 1- Fillard Pierre EPIDAURE/ASCLEPIOS Project-team, INRIA Sophia-Antipolis, 2004 Route des Lucioles BP 93, F-06902, Sophia Antipolis Cedex, France aut NATIONALLICENCE 700 1- Ayache Nicholas EPIDAURE/ASCLEPIOS Project-team, INRIA Sophia-Antipolis, 2004 Route des Lucioles BP 93, F-06902, Sophia Antipolis Cedex, France aut NATIONALLICENCE 773 0- International Journal of Computer Vision Kluwer Academic Publishers 66/1(2006-01-01), 41-66 0920-5691 66:1<41 2006 66 11263 Metadata rights reserved Springer special CC-BY-NC licence nationallicence BK010053 XK010053 XK010000 NATIONALLICENCE NATIONALLICENCE NL-springer