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A Riemannian Framework for Tensor Computing
Authors:Xavier Pennec  Pierre Fillard  Nicholas Ayache
Affiliation:(1) EPIDAURE/ASCLEPIOS Project-team, INRIA Sophia-Antipolis, 2004 Route des Lucioles BP 93, F-06902 Sophia Antipolis Cedex, France
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.
Keywords:tensors  diffusion tensor MRI  regularization  interpolation  extrapolation  PDE  Riemannian manifold  affine-invariant metric
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