3D stochastic completion fields for mapping connectivity in diffusion MRI

The 2D stochastic completion field algorithm, introduced by Williams and Jacobs [1], [2], uses a directional random walk to model the prior probability of completion curves in the plane. This construct has had a powerful impact in computer vision, where it has been used to compute the shapes of likely completion curves between edge fragments in visual imagery. Motivated by these developments, we extend the algorithm to 3D, using a spherical harmonics basis to achieve a rotation invariant computational solution to the Fokker-Planck equation describing the evolution of the probability density function underlying the model. This provides a principled way to compute 3D completion patterns and to derive connectivity measures for orientation data in 3D, as arises in 3D tracking, motion capture, and medical imaging. We demonstrate the utility of the approach for the particular case of diffusion magnetic resonance imaging, where we derive connectivity maps for synthetic data, on a physical phantom and on an in vivo high angular resolution diffusion image of a human brain.