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From Second to Higher Order Tensors in Diffusion-MRI

Aurobrata Ghosh 1 Rachid Deriche 1 
1 ODYSSEE - Computer and biological vision
DI-ENS - Département d'informatique - ENS Paris, CRISAM - Inria Sophia Antipolis - Méditerranée , ENS-PSL - École normale supérieure - Paris, Inria Paris-Rocquencourt, ENPC - École des Ponts ParisTech
Abstract : Diffusion MRI, which is sensitive to the Brownian motion of molecules, has become today an excellent medical tool for probing the tissue micro-structure of cerebral white matter in vivo and non-invasively. It makes it possible to reconstruct fiber pathways and segment major fiber bundles that reflect the structures in the brain which are not visible to other non-invasive imaging modalities. Since this is possible without operating on the subject, but by integrating partial information from Diffusion Weighted Images into a reconstructed 'complete' image of diffusion, Diffusion MRI opens a whole new domain of image processing. Here we shall explore the role that tensors play in the mathematical model. We shall primarily deal with Cartesian tensors and begin with 2nd order tensors, since these are at the core of Diffusion Tensor Imaging. We shall then explore higher and even ordered symmetric tensors, that can take into account more complex micro-geometries of biological tissues such as axonal crossings in the white matter.
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Aurobrata Ghosh, Rachid Deriche. From Second to Higher Order Tensors in Diffusion-MRI. Aja-Fernández, Santiago and de Luis García, Rodrigo and Tao, Dacheng and Li, Xuelong. Tensors in Image Processing and Computer Vision, Springer London, pp.315-334, 2009, Computer Science, 978-1-84882-298-6 (Print) 978-1-84882-299-3 (Online). ⟨10.1007/978-1-84882-299-3⟩. ⟨inria-00496858⟩

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