I am interested in the generalized Fourier transform of a measure $ \mu$ along a direction $\theta$, that is defined as follows: $$ (\mathcal{F}_g (\mu))(s\theta) := \int_{\mathbb{R}^d} e^{-i \pi g(x,s\theta)} d\mu(x), $$ where $s\in \mathbb{R}$, $\theta \in \mathbb{R}^p$, and the function $g$ represents the 'basis' of the transform. For instance, if $g(x,\theta) = x^\top \theta$, then we recover the classical Fourier transform.
My question is as follows: What are the required conditions for $g$ such that the resulting transform is injective?
I have tried to find references for finding the answer; however, I couldn't find any accessible reference. My current expectation is that $g$ should be homogeneous of order 1 in $\theta$.