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Orthogonal tensor invariants and the analysis of diffusion tensor magnetic resonance images
Author(s) -
Ennis Daniel B.,
Kindlmann Gordon
Publication year - 2006
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.20741
Subject(s) - anisotropy , diffusion mri , tensor (intrinsic definition) , isotropy , eigenvalues and eigenvectors , fractional anisotropy , tensor contraction , tensor field , mathematical analysis , invariant (physics) , mathematics , physics , geometry , exact solutions in general relativity , mathematical physics , optics , quantum mechanics , magnetic resonance imaging , medicine , radiology
Abstract This paper outlines the mathematical development and application of two analytically orthogonal tensor invariants sets. Diffusion tensors can be mathematically decomposed into shape and orientation information, determined by the eigenvalues and eigenvectors, respectively. The developments herein orthogonally decompose the tensor shape using a set of three orthogonal invariants that characterize the magnitude of isotropy, the magnitude of anisotropy, and the mode of anisotropy. The mode of anisotropy is useful for resolving whether a region of anisotropy is linear anisotropic, orthotropic, or planar anisotropic. Both tensor trace and fractional anisotropy are members of an orthogonal invariant set, but they do not belong to the same set. It is proven that tensor trace and fractional anisotropy are not mutually orthogonal measures of the diffusive process. The results are applied to the analysis and visualization of diffusion tensor magnetic resonance images of the brain in a healthy volunteer. The theoretical developments provide a method for generating scalar maps of the diffusion tensor data, including novel fractional anisotropy maps that are color encoded for the mode of anisotropy and directionally encoded colormaps of only linearly anisotropic structures, rather than of high fractional anisotropy structures. Magn Reson Med, 2006. © 2005 Wiley‐Liss, Inc.