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Summary: Fast and Simple Calculus on Tensors
in the Log-Euclidean Framework
Vincent Arsigny1
, Pierre Fillard1
, Xavier Pennec1
, and Nicholas Ayache1
INRIA Sophia - Projet Epidaure, BP 93, 06902 Sophia Antipolis Cedex, France
{Vincent.Arsigny, Pierre.Fillard, Xavier.Pennec,
Nicholas.Ayache}@Sophia.Inria.fr
Abstract. Computations on tensors have become common with the use
of DT-MRI. But the classical Euclidean framework has many defects,
and affine-invariant Riemannian metrics have been proposed to correct
them. These metrics have excellent theoretical properties but lead to
complex and slow algorithms. To remedy this limitation, we propose new
metrics called Log-Euclidean. They also have excellent theoretical prop-
erties and yield similar results in practice, but with much simpler and
faster computations. Indeed, Log-Euclidean computations are Euclidean
computations in the domain of matrix logarithms. Theoretical aspects
are presented and experimental results for multilinear interpolation and
regularization of tensor fields are shown on synthetic and real DTI data.
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