Rotation Invariant Completion Fields for Mapping Diffusion MRI Connectivity

Partial differential equations have been successfully used for fibre tractography and for mapping connectivity indices in the brain. However, the current implementation of methods which require 3D orientation to be tracked can suffer from serious shortcomings when invariance to 3D rotation is desired. In this paper we focus on the 3D stochastic completion field and introduce a new methodology to solve the underlying PDE in a manner that achieves rotation invariance. The key idea is to use spherical harmonics to solve the Fokker-Planck equation representing the evolution of the probability density function of a 3D directional random walk. We validate the new approach by presenting improved connectivity indices on synthetic data, on the MICCAI 2009 Fibre Cup phantom and on a biological phantom comprised of two rat spinal chords in a crossing configuration.