Sparse regularization for fiber ODF reconstruction: From the suboptimality of ℓ2ℓ2 and ℓ1ℓ1 priors to ℓ0ℓ0

• Sparse reconstruction methods have recently been proposed to shorten dMRI scan times.
• The L1 norm is normally used as a means to promote sparsity in the recovered FOD.
• Minimizing L1 is somehow conflicting with the fact that volume fractions sum to unity.
• We reformulate the problem using a L0 prior with the aim of optimally model sparsity.
• Our approach models more adequately the data and indeed improves the reconstructions.

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 ℓ2ℓ2-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 ℓ1ℓ1-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 ℓ1ℓ1-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 ℓ1ℓ1 inconsistency while simultaneously exploiting sparsity more optimally than through an ℓ2ℓ2 prior, we reformulate the reconstruction problem as a constrained formulation between a data term and a sparsity prior consisting in an explicit bound on the ℓ0ℓ0 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 ℓ0ℓ0 formulation significantly reduces modeling errors compared to the state-of-the-art ℓ2ℓ2 and ℓ1ℓ1 regularization approaches.

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