Mapping the local microstructure properties of the tissue in the brain is crucial to understand any pathological condition from a biological perspective. In recent years, microstructure-imaging techniques aimed at extracting such precious information by using explicit biophysical modeling of the decay patterns in different tissue compartments, e.g. axons, glial cells and extra-axonal space. In particular, the ActiveAx technique allows the estimation of the density and the average diameter of the axons with diffusion MRI, but the non-linear routines usually employed to fit the model to the data are computationally very intensive and cause practical problems for their application in clinical studies. Moreover, the model assumes a single axon orientation while numerous regions of the brain actually present more complex configurations, e.g. fiber crossing. ActiveAx was later extended to allow axon diameter mapping also in regions with crossing fibers, but the fitting time is still prohibitive for practical applications. Thus, there is an evident need for developing advanced techniques enabling fast and accurate reconstructions of the tissue microstructure properties not limited to regions known to contain only parallel fibers as for instance the corpus callosum. Recently, Daducci et al presented a flexible framework for Accelerated Microstructure Imaging via Convex Optimization (AMICO) to reformulate these microstructure imaging techniques as linear systems that can be solved using fast convex optimization methods. The purpose of the present study is to extend the AMICO framework, which in its current formulation assumes only one fiber population per voxel, to recover microstructure parameters also in regions with multiple fiber populations.