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