Orthogonal Tensor Invariants and the Analysis of Diffusion Tensor Magnetic Resonance Images

Daniel B. Ennis, G. Kindlmann
Magnetic Resonance in Medicine
Volume 55, Number 1, Pages 136-146
2006

Download full paper            DOI: 10.1002/mrm.20741

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.

Illustration of the two-dimensional space of diffusion tensor anisotropy, which can be parameterized with fractional anisotropy (FA=R2) and mode (R3=K3). Each glyph represents the shape of diffusion tensors with constant tensor norm rendered with superquadric glyphs. Increasing distance from the top left spherical glyph indicates increasing FA, whereas the angular deviation from the left edge indicates increasing mode, as it transitions from planar anisotropic (mode=-1), to ortho- tropic (mode=0), to linear anisotropic (mode=1). Glyphs along constant radii (constrained to an arc) are of constant fractional anisotropy, but of varying mode. This figure explicitly demonstrates that increases in FA do not necessarily indicate increasing linear anisotropy. The space of FA and mode is correctly diagrammed as a right triangle, this creates orthogonality between isocontours of mode and FA.

Reference

Ennis DB, Kindlmann G. Orthogonal tensor invariants and the analysis of diffusion tensor magnetic resonance images. Magnetic Resonance in Medicine 2006;55(1):136-146.

Bibtex entry

@article{ennisMRM06,
  author         = "Daniel B Ennis and Gordon Kindlmann",                      
  title          = "Orthogonal Tensor Invariants and the Analysis of  Diffusion
                   Tensor Magnetic Resonance Images",                          
  journal        = "Magnetic Resonance in Medicine",                           
  volume         = "55",                                                       
  number         = "1",                                                        
  pages          = "136--146",                                                 
  year           = "2006",                                                     
  doi            = "10.1002/mrm.20741"}

Grants

NIH T32-CA09695, NIH T32-EB002177

Copyright Information

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