  We give a generalization to bi-filtered $\mathcal D$-modules underlying mixed
Hodge modules of the relation between microlocalization along $f_1,...,f_r \in
\mathcal O_X(X)$ and vanishing cycles along $g = \sum_{i=1}^r y_i f_i$. This
leads to an interesting isomorphism between localization triangles. As an
application, we use these results to compare the $k$-plane Radon transform and
the Fourier-Laplace transform for mixed Hodge modules. This is then applied to
the Hodge module structure of certain GKZ systems.
