  We consider the Radon transform associated to dual pairs $(X,\Xi)$ in the
sense of Helgason, with $X=G/K$ and $\Xi=G/H$, where $G=\mathbb{R}^d\rtimes K$,
$K$ is a closed subgroup of ${\rm GL}(d,\mathbb{R})$ and $H$ is a closed
subgroup of $G$. Under some technical assumptions, we prove that if the quasi
regular representations of $G$ acting on $L^2(X)$ and $L^2(\Xi)$ are
irreducible, then the Radon transform admits a unitarization intertwining the
two representations. If, in addition, the representations are square
integrable, we provide an inversion formula for the Radon transform based on
the voice transform associated to these representations.
