In this study we developed a robust inversion algorithm to estimate the Neurite Orientation Dispersion and Density Imaging (NODDI) model. It is based on the Accelerated Microstructure Imaging via Convex Optimization (AMICO) framework. However, in contrast to AMICO, the proposed method relies on realistic MRI noise models. Moreover, it allows to take into account the underlying spatial continuity of the brain image by including a total variation regularization term. In simulated data the new method was effective in reducing the outliers, producing results more close to the ground-truth and with lower variability. The method was also evaluated on real data.