Let $\mathbb{S}^{N-1}$ be the unit sphere in $\mathbb{R}^N$ under the Euclidean norm $||\cdot||$. Let $\mu$ be an infinitely divisible Borel measure. If there exists a finite measure $\alpha$ on $\mathbb{S}^{N-1}$ with respect to $||\cdot||$ equipped with the Borel $\sigma$-algebra and a Borel function $\beta:\mathbb{S}^{N-1}\to\mathbb{R}_+$ such that the the Fourier transform of $\mu$ has the form $$\widehat{\mu}(\xi)=\exp\left(\int_{\mathbb{S}^{N-1}}\int_0^\infty \left(e^{ir\langle s,\xi\rangle}-1\right)\frac{e^{-\beta(s)r}}{r}dr\alpha(ds)\right)$$ for all $\xi\in\mathbb{R}^N$, then $\mu$ is called an $N$-dimensional Gamma distribution with parameters $\alpha,\beta$, and we write $\mu=\Gamma^N(\alpha,\beta)$.
My question is: how can $\mu$ be obtained from $\hat{\mu}$?