Using geodesics for inferring white matter fibre tracts from diffusion-weighted MR data is an attractive method for at least two reasons: (i) the method optimises a global criterion, and hence is less sensitive to local perturbations such as noise or partial volume effects, and (ii) the method is fast, allowing to infer on a large number of connexions in a reasonable computational time. Here, we propose an improved fast marching algorithm to infer on geodesic paths. Specifically, this procedure is designed to achieve accurate front propagation in an anisotropic elliptic medium, such as DTI data. We evaluate the numerical performance of this approach on simulated datasets, as well as its robustness to local perturbation induced by fiber crossing. On real data, we demonstrate the feasibility of extracting geodesics to connect an extended set of brain regions.