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Reports Year : 2004

A Riemannian Framework for Tensor Computing

Xavier Pennec
Pierre Fillard
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Nicholas Ayache


Positive definite symmetric matrices (so-called tensors in this article) are nowadays a common source of geometric information. In this paper, we propose to provide 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 manifold of constant curvature 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 schemes and Gaussian filtering 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 Laplacian 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 that are particularly simple and efficient to solve.


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Dates and versions

inria-00070743 , version 1 (19-05-2006)


  • HAL Id : inria-00070743 , version 1


Xavier Pennec, Pierre Fillard, Nicholas Ayache. A Riemannian Framework for Tensor Computing. RR-5255, INRIA. 2004, pp.34. ⟨inria-00070743⟩
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