Diffusion-weighted magnetic resonance imaging (D-MRI) is a technique to measure the diffusion of water, in biological tissues. It is used to detect microscopic patterns, such as neural fibers in the living human brain, with many medical and neuroscience applications e.g. for fiber tracking. In this paper, we consider High-Angular Resolution Diffusion Imaging (HARDI) which provides one of the richest representations of water diffusion. It records the movement of water molecules by measuring diffusion under 64 or more directions. A key challenge is that it generates high-dimensional, large, and complex datasets. In our work, we develop a novel representation that exploits the inherent sparsity of the HARDI signal by approximating it as a linear sum of basic atoms in an overcomplete data-driven dictionary using only a sparse set of coefficients. We show that this approach can be efficiently integrated into the standard q-ball imaging pipeline to compute the diffusion orientation distribution function (ODF). Sparse representations have the potential to reduce the size of the data while also giving some insight into the data. To explore the results, we provide a visualization of the atoms of the dictionary and their frequency in the data to highlight the basic characteristics of the data. We present our proposed pipeline and demonstrate its performance on 5 HARDI datasets.