A Riemannian Framework for Tensor Computing

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.
Complete list of metadatas

Cited literature [36 references]  Display  Hide  Download

https://hal.inria.fr/inria-00614990
Contributor : Project-Team Asclepios <>
Submitted on : Wednesday, August 17, 2011 - 9:11:39 PM
Last modification on : Monday, February 10, 2020 - 6:13:43 PM
Long-term archiving on: Friday, November 25, 2011 - 11:25:07 AM

File

Pennec.IJCV05.pdf
Files produced by the author(s)

Identifiers

Collections

Citation

Xavier Pennec, Pierre Fillard, Nicholas Ayache. A Riemannian Framework for Tensor Computing. International Journal of Computer Vision, Springer Verlag, 2006, 66 (1), pp.41--66. ⟨10.1007/s11263-005-3222-z⟩. ⟨inria-00614990⟩

Share

Metrics

Record views

558

Files downloads

635