Analytical computation of the eigenvalues and eigenvectors in DT-MRI.

In this paper a noniterative algorithm to be used for the analytical determination of the sorted eigenvalues and corresponding orthonormalized eigenvectors obtained by diffusion tensor magnetic resonance imaging (DT-MRI) is described. The algorithm uses the three invariants of the raw water spin self-diffusion tensor represented by a 3 x 3 positive definite matrix and certain math functions that do not require iteration. The implementation requires a positive definite mask to preserve the physical meaning of the eigenvalues. This algorithm can increase the speed of eigenvalue/eigenvector calculations by a factor of 5-40 over standard iterative Jacobi or singular-value decomposition techniques. This approach may accelerate the computation of eigenvalues, eigenvalue-dependent metrics, and eigenvectors especially when having high-resolution measurements with large numbers of slices and large fields of view.

[1]  Bahn.  Invariant and orthonormal scalar measures derived from magnetic resonance diffusion tensor imaging , 1999, Journal of magnetic resonance.

[2]  M. Bastin,et al.  A theoretical study of the effect of experimental noise on the measurement of anisotropy in diffusion imaging. , 1998, Magnetic resonance imaging.

[3]  E. Akbudak,et al.  Encoding of anisotropic diffusion with tetrahedral gradients: A general mathematical diffusion formalism and experimental results , 1996, Magnetic resonance in medicine.

[4]  Edmund R. Malinowski,et al.  Factor Analysis in Chemistry , 1980 .

[5]  P. Basser,et al.  A simplified method to measure the diffusion tensor from seven MR images , 1998, Magnetic resonance in medicine.

[6]  P J Basser,et al.  New Histological and Physiological Stains Derived from Diffusion‐Tensor MR Images , 1997, Annals of the New York Academy of Sciences.

[7]  D L Parker,et al.  Comparison of gradient encoding schemes for diffusion‐tensor MRI , 2001, Journal of magnetic resonance imaging : JMRI.

[8]  Victor J. Katz,et al.  A History of Mathematics: An Introduction , 1998 .

[9]  Eric W. Weisstein,et al.  The CRC concise encyclopedia of mathematics , 1999 .

[10]  M. Raichle,et al.  Tracking neuronal fiber pathways in the living human brain. , 1999, Proceedings of the National Academy of Sciences of the United States of America.

[11]  Donald E. Sands,et al.  Vectors and Tensors in Crystallography , 1982 .

[12]  G. Pearlson,et al.  In vivo visualization of human neural pathways by magnetic resonance imaging , 2000, Annals of neurology.

[13]  William H. Press,et al.  Numerical recipes in C , 2002 .

[14]  Peter L. Balise,et al.  Vector and Tensor Analysis with Applications , 1969 .

[15]  David Burton.  History of Mathematics an Introduction , 1988 .

[16]  Richard Bellman,et al.  Introduction to Matrix Analysis , 1972 .

[17]  P. Garnier,et al.  Evolution des tenseurs de dilatation thermique en fonction de la temperature. I. Loi generale d'evolution de la symetrie du tenseur , 1978 .

[18]  P. Basser,et al.  Toward a quantitative assessment of diffusion anisotropy , 1996, Magnetic resonance in medicine.

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

[20]  Society of magnetic resonance in medicine , 1990 .

[21]  D. Parker,et al.  A geometric analysis of diffusion tensor measurements of the human brain , 2000, Magnetic resonance in medicine.

[22]  P. V. van Zijl,et al.  Orientation‐independent diffusion imaging without tensor diagonalization: Anisotropy definitions based on physical attributes of the diffusion ellipsoid , 1999, Journal of magnetic resonance imaging : JMRI.

[23]  P. Basser,et al.  In vivo fiber tractography using DT‐MRI data , 2000, Magnetic resonance in medicine.

[24]  C. Poupon,et al.  Regularization of Diffusion-Based Direction Maps for the Tracking of Brain White Matter Fascicles , 2000, NeuroImage.

[25]  T. Mckeown.  Mechanics , 1970, The Mathematics of Fluid Flow Through Porous Media.