Directional detection is a promising Dark Matter search strategy. Indeed, WIMP-induced recoils present a direction dependence toward the Cygnus constellation, while background-induced recoils exhibit an isotropic distribution in the galactic rest frame. Taking advantage on these characteristic features and even in the presence of a sizeable background, we show for the first time the possibility to constrain the WIMP properties, both from particle and galactic halo physics, leading to an identification of non-baryonic Dark Matter. However, such results need highly accurate track reconstruction which should be reachable by the MIMAC detector using a dedicated readout combined with a likelihood analysis of recoiling nuclei.