Sparse regularization for fiber ODF reconstruction: from the suboptimality of $\ell_2$ and $\ell_1$ priors to $\ell_0$
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 $\ell_2$-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 $\ell_1$-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 $\ell_1$-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 $\ell_1$ inconsistency while simultaneously exploiting sparsity more optimally than through an $\ell_2$ 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 $\ell_0$ 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 $\ell_0$ formulation significantly reduces modeling errors compared to the state-of-the-art $\ell_2$ and $\ell_1$ regularization approaches.