From Second to Higher Order Tensors in Diffusion-MRI

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.

[1]  Zhizhou Wang,et al.  A constrained variational principle for direct estimation and smoothing of the diffusion tensor field from complex DWI , 2004, IEEE Transactions on Medical Imaging.

[2]  Nicholas Ayache,et al.  Fast and Simple Calculus on Tensors in the Log-Euclidean Framework , 2005, MICCAI.

[3]  Denis Le Bihan,et al.  Imagerie de diffusion in-vivo par résonance magnétique nucléaire , 1985 .

[4]  W. W. Hansen,et al.  Nuclear Induction , 2011 .

[5]  Peter J. Basser,et al.  Spectral decomposition of a 4th-order covariance tensor: Applications to diffusion tensor MRI , 2007, Signal Process..

[6]  E. Hahn,et al.  Spin Echoes , 2011 .

[7]  M. Moseley,et al.  Magnetic Resonance in Medicine 51:924–937 (2004) Characterizing Non-Gaussian Diffusion by Using Generalized Diffusion Tensors , 2022 .

[8]  Ludovico Minati,et al.  Physical foundations, models, and methods of diffusion magnetic resonance imaging of the brain: A review , 2007 .

[9]  T. Mareci,et al.  Generalized diffusion tensor imaging and analytical relationships between diffusion tensor imaging and high angular resolution diffusion imaging , 2003, Magnetic resonance in medicine.

[10]  P. Comon,et al.  A polynomial based approach to extract the maxima of an antipodally symmetric spherical function and its application to extract fiber directions from the Orientation Distribution Function in Diffusion MRI , 2008 .

[11]  Luke Bloy,et al.  On Computing the Underlying Fiber Directions from the Diffusion Orientation Distribution Function , 2008, MICCAI.

[12]  Rachid Deriche,et al.  Riemannian Framework for Estimating Symmetric Positive Definite 4th Order Diffusion Tensors , 2008, MICCAI.

[13]  Rachid Deriche,et al.  Statistics on the Manifold of Multivariate Normal Distributions: Theory and Application to Diffusion Tensor MRI Processing , 2006, Journal of Mathematical Imaging and Vision.

[14]  N. Ayache,et al.  Log‐Euclidean metrics for fast and simple calculus on diffusion tensors , 2006, Magnetic resonance in medicine.

[15]  Baba C. Vemuri,et al.  Symmetric Positive 4th Order Tensors & Their Estimation from Diffusion Weighted MRI , 2007, IPMI.

[16]  P. Basser,et al.  MR diffusion tensor spectroscopy and imaging. , 1994, Biophysical journal.

[17]  Xavier Pennec,et al.  A Riemannian Framework for Tensor Computing , 2005, International Journal of Computer Vision.

[18]  H. Pfeifer.  Principles of Nuclear Magnetic Resonance Microscopy , 1992 .

[19]  D. Tuch.  Diffusion MRI of complex tissue structure , 2002 .

[20]  Carl-Fredrik Westin,et al.  Processing and visualization for diffusion tensor MRI , 2002, Medical Image Anal..

[21]  E. Purcell,et al.  Resonance Absorption by Nuclear Magnetic Moments in a Solid , 1946 .

[22]  J. E. Tanner,et al.  Spin diffusion measurements : spin echoes in the presence of a time-dependent field gradient , 1965 .

[23]  Daniel Rueckert,et al.  Medical Image Computing and Computer-Assisted Intervention − MICCAI 2017: 20th International Conference, Quebec City, QC, Canada, September 11-13, 2017, Proceedings, Part II , 2017, Lecture Notes in Computer Science.

[24]  P. Mansfield.  Multi-planar image formation using NMR spin echoes , 1977 .

[25]  M C Bushell,et al.  PRELIMINARY COMMUNICATION: The spatial mapping of translational diffusion coefficients by the NMR imaging technique , 1985 .

[26]  V. Wedeen,et al.  Mapping fiber orientation spectra in cerebral white matter with Fourier-transform diffusion MRI , 2000 .

[27]  Klaus-Dietmarmerboldt.  Self-Diffusion NMR Imaging Using Stimulated Echoes , 2004 .

[28]  H. C. Torrey.  Bloch Equations with Diffusion Terms , 1956 .

[29]  P. Thomas Fletcher,et al.  Principal Geodesic Analysis on Symmetric Spaces: Statistics of Diffusion Tensors , 2004, ECCV Workshops CVAMIA and MMBIA.

[30]  Maher Moakher,et al.  A Differential Geometric Approach to the Geometric Mean of Symmetric Positive-Definite Matrices , 2005, SIAM J. Matrix Anal. Appl..

[31]  A. Einstein.  Über die von der molekularkinetischen Theorie der Wärme geforderte Bewegung von in ruhenden Flüssigkeiten suspendierten Teilchen [AdP 17, 549 (1905)] , 2005, Annalen der Physik.

[32]  J. Helpern,et al.  Diffusional kurtosis imaging: The quantification of non‐gaussian water diffusion by means of magnetic resonance imaging , 2005, Magnetic resonance in medicine.

[33]  E. Purcell,et al.  Effects of Diffusion on Free Precession in Nuclear Magnetic Resonance Experiments , 1954 .

[34]  D. Hilbert.  Über die Darstellung definiter Formen als Summe von Formenquadraten , 1888 .

[35]  P. Lauterbur,et al.  Image Formation by Induced Local Interactions: Examples Employing Nuclear Magnetic Resonance , 1973, Nature.

[36]  P. Basser,et al.  Estimation of the effective self-diffusion tensor from the NMR spin echo. , 1994, Journal of magnetic resonance. Series B.

[37]  Roland Bammer,et al.  Generalized Diffusion Tensor Imaging (GDTI): A Method for Characterizing and Imaging Diffusion Anisotropy Caused by Non‐Gaussian Diffusion , 2010 .

[38]  Christophe Lenglet,et al.  Geometric and variational methods for diffusion tensor MRI processing. (Méthodes géométriques et variationnelles pour le traitement d'IRM du tenseur de diffusion) , 2006 .

[39]  Rachid Deriche,et al.  Constrained Flows of Matrix-Valued Functions: Application to Diffusion Tensor Regularization , 2002, ECCV.