Finding frequency as a function of wavevector ( Fourier Space)

I have data for a function $$G(x,y,t)$$ where $$x,y$$ show space position (2D) and $$t$$ is time. I know from Fourier transformation definition that:

$$G(x,y,t)=\int q \:dq \: d\omega \: d\theta\: \hat{G}(q,\theta,\omega) e^{\omega t+i q(x\: cos\theta+y \:sin\theta)}$$

I want to use the data to find $$\omega$$ as a function of $$q$$ for different values of $$\theta$$ for example $$\theta=\pi/3$$ and $$q=0.1$$. Could somebody please help me in this regard?