Inferring White Matter Geometry from Di.usion Tensor MRI: Application to Connectivity Mapping

We introduce a novel approach to the cerebral white matter 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 complex 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 majors steps: First, we establish the link between Brownian motion and diffusion MRI by using the Laplace-Beltrami operator on M. We then expose how the sole knowledge 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 propose 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.