MAPL1: q-space reconstruction using ℓ1 -regularized mean apparent propagator

Abstract

Purpose: To improve the quality of mean apparent propagator (MAP) reconstruction from a limited number of q-space samples.

Methods: We implement an ${\ell }_1$ -regularised MAP (MAPL1) to consider higher order basis functions and to improve the fit without increasing the number of q-space samples. We compare MAPL1 with the least-squares optimization subject to non-negativity (MAP), and the Laplacian-regularized MAP (MAPL). We use simulations of crossing fibers and compute the normalized mean squared error (NMSE) and the Pearson's correlation coefficient to evaluate the reconstruction quality in q-space. We also compare coefficient-based diffusion indices in the simulations and in in vivo data.

Results: Results indicate that MAPL1 improves NMSE in 1 to 3% when compared to MAP or MAPL in a high undersampling regime. Additionally, MAPL1 produces more reproducible and accurate results for all sampling rates when there are enough basis functions to meet the sparsity criterion for the regularizer. These improved reconstructions also produce better coefficient-based diffusion indices for in vivo data.

Conclusions: Adding an ${\ell }_1$ regularizer to MAP allows the use of more basis functions and a better fit without increasing the number of q-space samples. The impact of our research is that a complete diffusion spectrum can be reconstructed from an acquisition time very similar to a diffusion tensor imaging protocol.

Keywords: Laplacian-regularizer; compressed sensing; diffusion magnetic resonance imaging; diffusion propagator; mean apparent propagator; q-space reconstruction.