Diffusion magnetic resonance imaging (dMRI) is a non-invasive method that allows connectivity mapping of the brain. However, despite major advances in this field, accurate inference of these patterns and its applicability within a clinical context is still in its early stages. This thesis describes a conceptually novel method for reconstructing neuronal pathways inside the brain from diffusion-weighted imaging (DWI) measurements with high angular resolution and short data acquisition time. The proposed method combines recent theoretical advances on spherical sampling and noise reduction techniques from the field of compressed sensing. Numerical simulations were performed to study the best sampling strategy under a novel sampling theorem on the sphere in order to reduce the acquisition time during dMRI scans. Furthermore, these results were combined with the recently proposed spherical deconvolution technique to reconstruct the distribution of neuronal tracts (or fibers) within one voxel with high angular resolution between multiple crossing fibers. The spherical deconvolution problem was hereby formulated as an inverse problem and solved using techniques adopted from the field of compressed sensing. Since the result of the spherical deconvolution step is sparse in nature, the basis pursuit denoising formulation of the inverse problem is optimal within this context. Finally, the resulting fiber orientation reconstruction was compared with diffusion spectrum imaging (DSI) – a classic model-free acquisition method. Simulations revealed that the proposed approach is superior to DSI in terms of both, acquisition time and angular resolution of crossing fibers (>=40° with at least 90% sensitivity). Our investigations showed that the application of spherical deconvolution stated as a basis pursuit denoising problem holds great promise for high angular resolution dMRI.