  Diffusion MRI is a well established imaging modality providing a powerful way
to probe the structure of the white matter non-invasively. Despite its
potential, the intrinsic long scan times of these sequences have hampered their
use in clinical practice. For this reason, a large variety of methods have been
recently proposed to shorten the acquisition times. Among them, spherical
deconvolution approaches have gained a lot of interest for their ability to
reliably recover the intra-voxel fiber configuration with a relatively small
number of data samples. To overcome the intrinsic instabilities of
deconvolution, these methods use regularization schemes generally based on the
assumption that the fiber orientation distribution (FOD) to be recovered in
each voxel is sparse. The well known Constrained Spherical Deconvolution (CSD)
approach resorts to Tikhonov regularization, based on an l2-norm prior, which
promotes a weak version of sparsity. Also, in the last few years compressed
sensing has been advocated to further accelerate the acquisitions and l1-norm
minimization is generally employed as a means to promote sparsity in the
recovered FODs. In this paper, we provide evidence that the use of an l1-norm
prior to regularize this class of problems is somewhat inconsistent with the
fact that the fiber compartments all sum up to unity. To overcome this l1
inconsistency while simultaneously exploiting sparsity more optimally than
through an l2 prior, we reformulate the reconstruction problem as a constrained
formulation between a data term and and a sparsity prior consisting in an
explicit bound on the l0 norm of the FOD, i.e. on the number of fibers. The
method has been tested both on synthetic and real data. Experimental results
show that the proposed l0 formulation significantly reduces modeling errors
compared to the state-of-the-art l2 and l1 regularization approaches.
