State-of-the-art microstructure imaging methods usually fit biophysical models to the diffusion MRI data on a voxel-by-voxel basis using non-linear procedures that require both long acquisitions and processing time. We recently introduced AMICO, a framework to reformulate these techniques as efficient linear problems and enable faster reconstructions. Here, we propose an extension that enables robust reconstructions from a reduced number of diffusion measurements, thus leading to faster acquisitions, too. Our novel formulation estimates simultaneously the microstructure configuration in all voxels as a global optimization problem, exploiting information from neighboring voxels that cannot be taken into account with existing techniques.