Skip to Main content Skip to Navigation

Fast and Simple Computations on Tensors with Log-Euclidean Metrics.

Vincent Arsigny 1 Pierre Fillard 1 Xavier Pennec 1 Nicholas Ayache 1
1 EPIDAURE - Medical imaging and robotics
CRISAM - Inria Sophia Antipolis - Méditerranée
Abstract : Computations on tensors, i.e. symmetric positive definite real matrices in medical imaging, appear in many contexts. In medical imaging, these computations have become common with the use of DT-MRI. The classical Euclidean framework for tensor computing has many defects, which has recently led to the use of Riemannian metrics as an alternative. So far, only affine-invariant metrics had been proposed, which have excellent theoretical properites but lead to complex algorithms with a high computational cost. In this article, we present a new familly of metrics, called Log-Euclidean. These metrics have the same excellent theoretical properties as affine-invariant metrics and yield very similar results in practice. But they lead to much more simple computations, with a much lighter computational cost, very close to the cost of the classical Euclidean framework. Indeed, Riemannian computations become Euclidean computations in the logarithmic domain with Log-Euclidean metrics. We present in this article the complete theory for these metrics, and show experimental results for multilinear interpolation, dense extrapolation of tensors and anisotropic diffusion of tensor fields.
Document type :
Complete list of metadatas

Cited literature [1 references]  Display  Hide  Download
Contributor : Rapport de Recherche Inria <>
Submitted on : Friday, May 19, 2006 - 8:27:00 PM
Last modification on : Saturday, January 27, 2018 - 1:31:27 AM
Document(s) archivé(s) le : Sunday, April 4, 2010 - 9:11:01 PM


  • HAL Id : inria-00070423, version 1



Vincent Arsigny, Pierre Fillard, Xavier Pennec, Nicholas Ayache. Fast and Simple Computations on Tensors with Log-Euclidean Metrics.. [Research Report] RR-5584, INRIA. 2005, pp.42. ⟨inria-00070423⟩



Record views


Files downloads