High Angular Resolution Diffusion Imaging (HARDI) demands a higher amount of data measurements compared to Diffusion Tensor Imaging (DTI), restricting its use in practice. We propose to represent the probabilistic Orientation Distribution Function (ODF) in the frame of Spherical Wavelets (SW), where it is highly sparse. From a reduced subset of measurements (nearly four times less than the standard for HARDI), we pose the estimation as an inverse problem with sparsity regularization. This allows the fast computation of a positive, unit-mass, probabilistic ODF from 14-16 samples, as we show with both synthetic diffusion signals and real HARDI data with typical parameters.