# Riemannian processing of tensors for diffusion MRI and computational anatomy of the brain.

1 ASCLEPIOS - Analysis and Simulation of Biomedical Images
CRISAM - Inria Sophia Antipolis - Méditerranée
Abstract : Symmetric, positive-definite matrices, or tensors, are nowadays a common geometrical tool for image processing and analysis. The recent emergence of diffusion tensor MRI (DTI) and computational anatomy (CA) brought importance of tensors out to the medical community. However, working with those is difficult: the positive-definite constraint must be satisfied at any cost, which cannot be ensured in general with standard matrix operations. In this work, we propose two alternatives to the standard Euclidean calculus on tensors. Instead of seeing the tensor space as a vector space, we consider it as a manifold, i.e., a smooth curved space. Thanks to the Riemannian geometry, we are able to unfold'' this space, and to generalize any operation on tensors with astonishing simple implementations. In a second step, we review the applications of such frameworks in the context of clinical DTI and brain CA. In DTI, we show that very noisy data, typical of clinical acquisitions, can be optimally exploited and eventually produce a meaningful and clinically relevant fiber reconstruction. In brain CA, we show that, by considering simple brain anatomical landmarks - the sulcal lines - we are able to precisely measure the inter-individual variability of the cortex. Finally, we develop a new framework to study the anatomical correlations between brain regions, and present results of so far unknown relationships between symmetric sulcal positions, and between a-priori unrelated sulci, which raises new fundamental questions about the origin of such statistical dependencies.
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Cited literature [181 references]

https://tel.archives-ouvertes.fr/tel-00265129
Contributor : Pierre Fillard <>
Submitted on : Tuesday, March 18, 2008 - 3:24:19 PM
Last modification on : Friday, January 18, 2019 - 1:19:54 AM
Long-term archiving on: : Thursday, May 20, 2010 - 8:12:07 PM

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• HAL Id : tel-00265129, version 1

### Citation

Pierre Fillard. Riemannian processing of tensors for diffusion MRI and computational anatomy of the brain.. Human-Computer Interaction [cs.HC]. Université Nice Sophia Antipolis, 2008. English. ⟨tel-00265129⟩

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