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.