Fiber Orientation Distribution Estimation Using a Peaceman-Rachford Splitting Method

In diffusion-weighted magnetic resonance imaging, the estimation of the orientations of multiple nerve fibers in each voxel (the fiber orientation distribution (FOD)) is a critical issue for exploring the connection of cerebral tissue. In this paper, we establish a convex semidefinite programming (CSDP) model for the FOD estimation. One feature of the new model is that it can ensure the statistical meaning of FOD since as a probability density function, FOD must be nonnegative and have a unit mass. To construct such a statistically meaningful FOD, we consider its approximation by a sum of squares (SOS) polynomial and impose the unit-mass by a linear constraint. Another feature of the new model is that it introduces a new regularization based on the sparsity of nerve fibers. Due to the sparsity of the orientations of nerve fibers in cerebral white matter, a heuristic regularization is raised, which is inspired by the Z-eigenvalue of a symmetric tensor that closely relates to the SOS polynomial. To solve the CSDP efficiently, we propose a new Peaceman-Rachford splitting method and prove its global convergence. Numerical experiments on synthetic and real-world human brain data show that, when compared with some existing approaches for fiber estimations, the new method gives a sharp and smooth FOD. Further, the proposed Peaceman-Rachford splitting method is shown to have good numerical performances comparing several existing methods.