Determination of axonal pathways provides an invaluable means to study the connectivity of the human brain and its functional network. Diffusion tensor imaging (DTI) is unique in its ability to capture the restricted diffusion of water molecules which can be used to infer the directionality of tissue components. In this paper, we introduce a white matter tractography method based on anisotropic wavefront propagation in diffusion tensor images. A front propagates in the white matter with a speed profile governed by the isocontour of the diffusion tensor ellipsoid. By using the ellipsoid, we avoid possible misclassification of the principal eigenvector in oblate regions. The wavefront evolution is described by an anisotropic version of the static Hamilton-Jacobi equation, which is solved by a sweeping method in order to obtain correct arrival times. Pathways of connection are determined by tracing minimum-cost trajectories using the characteristic vector field of the resulting partial differential equation. A validity index is described to rate the goodness of the resulting pathways with respect to the directionality of the tensor field. Connectivity results using normal human DTI brain images are illustrated and discussed. We also compared our method with a similar level set-based tractography technique, and found that the anisotropic evolution increased the validity index of the obtained pathways by 18%.