On Computing the Underlying Fiber Directions from the Diffusion Orientation Distribution Function

In this work, a novel method for determining the principal directions (maxima) of the diffusion orientation distribution function (ODF) is proposed. We represent the ODF as a symmetric high-order Cartesian tensor restricted to the unit sphere and show that the extrema of the ODF are solutions to a system of polynomial equations whose coefficients are polynomial functions of the tensor elements. In addition to demonstrating the ability of our methods to identify the principal directions in real data, we show that this method correctly identifies the principal directions under a range of noise levels. We also propose the use of the principal curvatures of the graph of the ODF function as a measure of the degree of diffusion anisotropy in that direction. We present simulated results illustrating the relationship between the mean principal curvature, measured at the maxima, and the fractional anisotropy of the underlying diffusion tensor.