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Diffusion MRI is a well established imaging modality providing a powerful wayto probe the structure of the white matter non-invasively. Despite itspotential, the intrinsic long scan times of these sequences have hampered theiruse in clinical practice. For this reason, a large variety of methods have beenrecently proposed to shorten the acquisition times. Among them, sphericaldeconvolution approaches have gained a lot of interest for their ability toreliably recover the intra-voxel fiber configuration with a relatively smallnumber of data samples. To overcome the intrinsic instabilities ofdeconvolution, these methods use regularization schemes generally based on theassumption that the fiber orientation distribution (FOD) to be recovered ineach voxel is sparse. The well known Constrained Spherical Deconvolution (CSD)approach resorts to Tikhonov regularization, based on an l2-norm prior, whichpromotes a weak version of sparsity. Also, in the last few years compressedsensing has been advocated to further accelerate the acquisitions and l1-normminimization is generally employed as a means to promote sparsity in therecovered FODs. In this paper, we provide evidence that the use of an l1-normprior to regularize this class of problems is somewhat inconsistent with thefact that the fiber compartments all sum up to unity. To overcome this l1inconsistency while simultaneously exploiting sparsity more optimally thanthrough an l2 prior, we reformulate the reconstruction problem as a constrainedformulation between a data term and and a sparsity prior consisting in anexplicit bound on the l0 norm of the FOD, i.e. on the number of fibers. Themethod has been tested both on synthetic and real data. Experimental resultsshow that the proposed l0 formulation significantly reduces modeling errorscompared to the state-of-the-art l2 and l1 regularization approaches.

Sparse regularization for fiber ODF reconstruction: from the suboptimality of $\ell_2$ and $\ell_1$ priors to $\ell_0$