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A rigorous framework for diffusion tensor calculus
Author(s) -
Batchelor P. G.,
Moakher M.,
Atkinson D.,
Calamante F.,
Connelly A.
Publication year - 2005
Publication title -
magnetic resonance in medicine
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.696
H-Index - 225
eISSN - 1522-2594
pISSN - 0740-3194
DOI - 10.1002/mrm.20334
Subject(s) - diffusion mri , tensor (intrinsic definition) , positive definiteness , geodesic , mathematics , eigenvalues and eigenvectors , tensor density , measure (data warehouse) , tensor contraction , cartesian tensor , symmetric tensor , tensor calculus , space (punctuation) , mathematical analysis , diffusion , tensor field , pure mathematics , exact solutions in general relativity , positive definite matrix , computer science , physics , medicine , quantum mechanics , database , magnetic resonance imaging , radiology , operating system , thermodynamics
In biological tissue, all eigenvalues of the diffusion tensor are assumed to be positive. Calculations in diffusion tensor MRI generally do not take into account this positive definiteness property of the tensor. Here, the space of positive definite tensors is used to construct a framework for diffusion tensor analysis. The method defines a distance function between a pair of tensors and the associated shortest path (geodesic) joining them. From this distance a method for computing tensor means, a new measure of anisotropy, and a method for tensor interpolation are derived. The method is illustrated using simulated and in vivo data. Magn Reson Med 53:221–225, 2005. © 2004 Wiley‐Liss, Inc.