Abstract

We introduce a novel approach to the cerebral white mat- ter connectivity mapping from diffusion tensor MRI. DT-MRI is the unique non-invasive technique capable of probing and quantifying the anisotropic diffusion of water molecules in biological tissues. We address the problem of consistent neural fibers reconstruction in areas of com- plex diffusion profiles with potentially multiple fibers orientations. Our method relies on a global modelization of the acquired MRI volume as a Riemannian manifold M and proceeds in 4 ma jors steps: First, we estab- lish the link between Brownian motion and diffusion MRI by using the Laplace-Beltrami operator on M . We then expose how the sole knowl- edge of the diffusion properties of water molecules on M is sufficient to infer its geometry. There exists a direct mapping between the diffusion tensor and the metric of M . Next, having access to that metric, we pro- pose a novel level set formulation scheme to approximate the distance function related to a radial Brownian motion on M . Finally, a rigorous numerical scheme using the exponential map is derived to estimate the geodesics of M , seen as the diffusion paths of water molecules. Numerical experimentations conducted on synthetic and real diffusion MRI datasets illustrate the potentialities of this global approach.