Recent development in sampling theory now allows the sampling and reconstruction of certain non-bandlimited functions on the sphere, namely a sum of weighted Diracs. Because the signal acquired in diffusion Magnetic Resonance Imaging (dMRI) can be modeled as the convolution between a sampling kernel and two dimensional Diracs defined on the sphere, these advances have great potential in dMRI. In this work, we introduce a local reconstruction method for dMRI based on a new sampling theorem for non-bandlimited signals on the sphere. This new algorithm, named Spherical Finite Rate of Innovation (SFRI), is able to recover fibers crossing at very narrow angles with little dependence on the b-value. Because of its parametric formulation, SFRI can distinguish crossing fibers even when using a DTI-like acquisition (≈32 directions). This opens new perspective for low b-value and low number of gradient directions diffusion acquisitions and tractography studies. We evaluate the angular resolution of SFRI using state of the art synthetic data and compare its performance using in-vivo data. Our results show that, at low b-values, SFRI recovers crossing fibers not identified by constrained spherical deconvolution. We also show that low b-value results obtained using SFRI are similar to those obtained with constrained spherical deconvolution at a higher b-value.