Supplemental Document for “Fiber Direction Estimation, Smoothing and Tracking in Diffusion MRI”

This document provides supplementary material to the article “Fiber Direction Estimation, Smoothing and Tracking in Diffusion MRI” written by the same authors. S1 Practical maximum likelihood estimation for model (2) In an attempt to find the global maximizer of the likelihood (3), we develop an efficient algorithm through an approximation of model (2). This algorithm essentially performs a grid search, but it makes use of the geometry of the problem so it is fast. It includes three major steps: (i) lay down a grid for (αj ,m ⊺ j )’s, (ii) evaluate the maximized likelihood function w.r.t. τj ’s on the grid, and (iii) return the grid point that maximizes the likelihood function. One can then use this returned grid point as a starting value in a gradient method for obtaining ML estimation of model (2). Such a strategy results in better numerical stability and accuracy in finding ML estimates. S1.1 An approximation of model (2) Let cj = (αj ,m ⊺ j ) ⊺, c = (c1, . . . , c ⊺ J) ⊺ and Cj be the set of grid points for cj . For simplicity, we take the same set of grid points, C, for all j. To lay down a grid for mj ’s, we apply the sphere tessellation using Icosahedron, which is depicted in Figure S1. The tessellation algorithm starts