Microlocal analysis in generalized function algebras based on generalized points and generalized directions

We develop a refined theory of microlocal analysis in the algebra ${\mathcal G}(\Omega)$ of Colombeau generalized functions. In our approach, the wave front is a set of generalized points in the cotangent bundle of $\mathrm{\Omega }$, whereas in the theory developed so far, it is a set of nongeneralized points. We also show consistency between both approaches.