  We present a novel method for estimation of the fiber orientation
distribution (FOD) function based on diffusion-weighted Magnetic Resonance
Imaging (D-MRI) data. We formulate the problem of FOD estimation as a
regression problem through spherical deconvolution and a sparse representation
of the FOD by a spherical needlets basis that form a multi-resolution tight
frame for spherical functions. This sparse representation allows us to estimate
FOD by an $l_1$-penalized regression under a non-negativity constraint. The
resulting convex optimization problem is solved by an alternating direction
method of multipliers (ADMM) algorithm. The proposed method leads to a
reconstruction of the FODs that is accurate, has low variability and preserves
sharp features. Through extensive experiments, we demonstrate the effectiveness
and favorable performance of the proposed method compared with two existing
methods. Particularly, we show the ability of the proposed method in
successfully resolving fiber crossing at small angles and in automatically
identifying isotropic diffusion. We also apply the proposed method to real 3T
D-MRI data sets of healthy elderly individuals. The results show realistic
descriptions of crossing fibers that are more accurate and less noisy than
competing methods even with a relatively small number of gradient directions.
