Sparse regularization for fiber ODF reconstruction: From the suboptimality of ℓ2 and ℓ1 priors to ℓ0

Alessandro Daducci*, Dimitri Van De Ville, Jean-Philippe Thiran, Yves Wiaux

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

48 Citations (Scopus)
32 Downloads (Pure)


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 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.

Original languageEnglish
Pages (from-to)820-833
Number of pages14
JournalMedical Image Analysis
Issue number6
Publication statusPublished - Aug 2014


  • Compressed sensing
  • Diffusion MRI
  • Reconstruction

ASJC Scopus subject areas

  • Computer Graphics and Computer-Aided Design
  • Computer Vision and Pattern Recognition
  • Radiology Nuclear Medicine and imaging
  • Health Informatics
  • Radiological and Ultrasound Technology


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