Exact integration of diffusion orientation distribution functions for graph-based diffusion MRI analysis

Graph-based image analysis methods are increasingly being applied to diffusion MRI (dMRI) analysis. Unfortunately, weighting the graph for these methods involves solving a complex integral of an orientation distribution function (ODF). To date, these integrals have been approximated numerically at high computational cost and have resulted in numerical approximation errors that degrade dMRI analysis results. By exploiting a spherical harmonic representation of the ODF, we derive for the first time an analytical solution to the edge weight integrals used in graph-based dMRI analysis. We further show that the computational efficiency of our analytical integration is over forty times faster than numerical approximation schemes on typical data sets. Further, we incorporate our exact integration scheme into an existing graph-based probabilistic tractography method and show a reduction in error accumulation in the resulting tractograms.