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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.
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https://hal.inria.fr/inria-00070423
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Submitted on : Friday, May 19, 2006 - 8:27:00 PM
Last modification on : Saturday, January 27, 2018 - 1:31:27 AM
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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⟩

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