My professor for a computer vision class introduced the 2-D Fourier Transform and Inverse Fourier Transform (in the context of images) as the following:
$$X(\omega_1, \omega_2)=\sum_{n_1}\sum_{n_2}x(n_1,n_2)e^{j\omega_1n_1}e^{j\omega_2n_2}$$
and
$$x(n_1,n_2)=\frac{1}{4\pi^2}\int_{-\pi}^{\pi}\int_{-\pi}^{\pi}X(\omega_1,\omega_2)e^{j\omega_1n_1}e^{j\omega_2n_2}d\omega_1 d\omega_2,$$
where $x$ is the original image and $X$ is the Fourier Transform of $x$.
In the IFT, there are bounds from $-\pi$ to $\pi$, and I'm not seeing why this is the case; it seems like to me they should be from $-\infty$ to $\infty$, because in the FT formula, $\omega_1$ and $\omega_2$ can take on any value, so we need to integrate over all those values. I was told that the complex exponential in the integral is periodic, so we don't need to integrate over the entire space. This doesn't seem like a complete explanation to me — I think it's because $X(\omega_1,\omega_2)$ is also periodic? (Please confirm whether this is true.) In that case, the bounds would make sense since the period seems to be $2\pi$.
If it is true, I was wondering why most versions of the 2-D FT/IFT that I've found on the internet integrate from $-\infty,+\infty$.