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A Riemannian Framework for the Processing of Tensor-Valued Images

Summary: A Riemannian Framework for the Processing of
Tensor-Valued Images
Pierre Fillard, Vincent Arsigny, Nicholas Ayache, and Xavier Pennec
INRIA Sophia Antipolis - Epidaure Project,
2004 Route des Lucioles BP 93,
06902 Sophia Antipolis Cedex, France
{Pierre.Fillard, Vincent.Arsigny, Nicholas.Ayache,
Abstract. In this paper, we present a novel framework to carry out
computations on tensors, i.e. symmetric positive definite matrices. We
endow the space of tensors with an affine-invariant Riemannian metric,
which leads to strong theoretical properties: The space of positive definite
symmetric matrices is replaced by a regular and geodesically complete
manifold without boundaries. Thus, tensors with non-positive eigenval-
ues are at an infinite distance of any positive definite matrix. Moreover,
the tools of differential geometry apply and we generalize to tensors nu-
merous algorithms that were reserved to vector spaces. The application
of this framework to the processing of diffusion tensor images shows
very promising results. We apply this framework to the processing of
structure tensor images and show that it could help to extract low-level


Source: Ayache, Nicholas - INRIA


Collections: Computer Technologies and Information Sciences; Engineering