Formulation and Efficient Computation of ${\ell}_{\textsf{1}}$ - and Smoothness Penalized Estimates for Microstructure-Informed Tractography

Fiber tractography based on diffusion-weighted magnetic resonance imaging is to date the only method for the three-dimensional visualization of nerve fiber bundles in the living human brain noninvasively. However, various existing methods suffer from reconstructing anatomically implausible fiber tracks due to exclusive local treatment of the input data. A method that seeks to filter out invalid tracks in a postprocessing step by solving a convex optimization problem with l
₁
-norm regularization was recently introduced in the work by Daducci et al. In this paper, we derive an improved version of this method by adding Sobolev-norm regularization terms. Furthermore, we present a robust and efficient strategy using the alternating direction method of multipliers and dimension reduction using truncated singular value decomposition to solve the resulting optimization problem. The qualitative results show the applicability of the algorithm to large in vivo data sets. The quantitative results and numerical experiments for diffusion phantom data with known ground truth show the benefits of the proposed method.