Positive semidefinite generalized diffusion tensor imaging via quadratic semidefinite programming

The positive definiteness of a diffusion tensor is important in magnetic resonance imaging because it reflects the phenomenon of water molecular diffusion in complicated biological tissue environments. To preserve this property, we represent it as an explicit positive semidefinite (PSD) matrix constraint and some linear matrix equalities. The objective function is the regularized linear least squares fitting for the log-linearized Stejskal-Tanner equation. The regularization term is the heuristic nuclear norm of the PSD matrix, since we expect it to be of low rank. In this way, we establish a convex quadratic semidefinite programming (SDP) model, whose global solution exists. The optimal solution could be solved by three efficient methods. While there are two state-of-the-art solvers-SDPT3 and QSDP- for the primal problem, we design a new augmented Lagrangian based alternating direction method (ADM) for the dual problem. Sensitivity analyses on the coefficients of the optimal diffusion tensor and the optimal objective function value with respect to noise-corrupted signals are presented. Experiments on synthetic data with multiple fibers show that the new method is robust to the Rician noise and outperforms several existing methods. Furthermore, when the fiber orientation distribution function is considered, the new method is competitive with the Q-ball imaging. Using the human brain data, we illustrate that the new method could capture the crossing of three nervous fiber bundles. Additionally, the new method generates positive definite generalized diffusion tensors in all voxels, while the unconstrained least squares fitting fails. Finally, we confirm that the ADM solver is more efficient than SDPT3 and QSDP for this special problem.