We propose a novel variational framework for the dense non-rigid registration of Diffusion Tensor Images (DTI). Our approach relies on the differential geometrical properties of the Riemannian manifold of multivariate normal distributions endowed with the metric derived from the Fisher information matrix. The availability of closed form expressions for the geodesics and the Christoffel symbols allows us to define statistical quantities and to perform the parallel transport of tangent vectors in this space. We propose a matching energy that aims to minimize the difference in the local statistical content (means and covariance matrices) of two DT images through a gradient descent procedure. The result of the algorithm is a dense vector field that can be used to wrap the source image into the target image. This article is essentially a mathematical study of the registration problem. Some numerical experiments are provided as a proof of concept
