Registration of Diffusion-weighted imaging (DWI) data emerges as an important topic in magnetic resonance (MR) image analysis. As existing methods are often designed for specific diffusion models, it is difficult to fit to the registered data different models other than the one used for registration. In this paper we describe a diffeomorphic registration algorithm for DWI data in a large deformation setting. Our method generates spatially normalized DWI data and it is thus possible to fit various diffusion models after registration for comparison purposes. Our algorithm includes (1) a reorientation component, where each diffusion profile (DWI signal as a function on a unit sphere) is decomposed, reoriented and recomposed to form the orientation-corrected DWI profile, and (2) a large deformation diffeomorphic registration component to ensure one-to-one mapping in a large-structural-variation scenario. In addition our algorithm uses a geodesic shooting mechanism to avoid the huge computational resources that are needed to register high-dimensional vector-valued data. We also incorporate into our algorithm a multi-kernel strategy where anatomical structures at different scales are considered simultaneously during registration. We demonstrate the efficacy of our method using in vivo data.