The Hamilton-Jacobi equation (HJE) appears widely in applied mathematics, physics, and optimal control
theory. While its analytical solution is rarely available, the numerical solver is indispensable. In this
work, firstly we propose a novel numerical method, based on the fast sweeping scheme, for the static HJE.
Comparing with the original fast sweeping method, our algorithm speeds up the solution up to 8 times in
3D. The efficiency is due to incorporating the ideas of the fast marching into the fast sweeping. Essentially,
the sweeping origin is selected so that the sweeping direction is more consistent with the information flow
direction and the regions where the two directions are against are avoided. Moreover, the successive-overrelaxation
nonlinear iterative method is used for faster convergence. Secondly, we provide a complete pipeline
for brain tractography, in which the proposed solver is the key component for finding the optimal fiber tracts.
Besides, the pipeline contains components from orientation distribution function estimation, multiple fiber
extraction to the final fiber bundle volumetric segmentation, completing the process from DW-MRI image to segmented fiber bundles. The pipeline is integrated into the publicly available software 3D Slicer. The new solver has been tested and compared with the original scheme on various types of HJEs and the tractography
pipeline was tested and performed consistently on all the 12 brain DW-MRI images.