IntraVoxel Incoherent Motion (IVIM) Diffusion-Weighted Magnetic Resonance Imaging (DW-MRI) is of great interest for evaluating tissue diffusion and perfusion and producing parametric maps in clinical applications for liver pathologies. However, the presence of macroscopic blood vessels (not capillaries) in a given Region of Interest (ROI) results in a confounding effect that bias the quantification of tissue perfusion. Therefore, it is necessary to identify those voxels affected by blood vessels. In this paper, an efficient algorithm for an automatic identification of blood vessels in a given ROI is proposed. It relies on the sparsity of the spatial distribution of blood vessels. This sparsity prior can be easily incorporated using the all-voxel IVIM-MRI model introduced in this paper. In addition to the identification of blood vessels, the proposed algorithm provides a quantification of blood vessels, tissue diffusion and tissue perfusion of all voxels in a given ROI, in one single step. Besides, two strategies are proposed in this paper to deal with the nonnegativity of the model parameters. The efficiency of the proposed algorithm compared to the Non-Negative Least Square (NNLS)-based method, recently introduced to deal with the confounding blood vessel effect in the IVIM-MRI model, is confirmed using both realistic and real DW-MR images.