12 March 2010 Fast Hamilton-Jacobi equation solver and neural fiber bundle extraction
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Abstract
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.
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Yi Gao, Yi Gao, John Melonakos, John Melonakos, Allen Tannenbaum, Allen Tannenbaum, } "Fast Hamilton-Jacobi equation solver and neural fiber bundle extraction", Proc. SPIE 7623, Medical Imaging 2010: Image Processing, 76231F (12 March 2010); doi: 10.1117/12.840710; https://doi.org/10.1117/12.840710
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