Constructing a Gaussian kernel in the frequency domain I'm currently learning about Fourier transform, but find the differences between spatial domain and frequency domain a bit confusing at times.
Let's say I would like to perform convolution of an image with a Gaussian kernel. As far as i understand the Fourier transform of a Gaussian is also a Gaussian i.e the Fourier transform of
$$
g(x,y;\sigma) = \frac{1}{2\pi\sigma^2} \cdot e^{- \frac{x^2+y^2}{2\sigma^2}}
$$
is
$$
G(x,y;\sigma) = e^{- 2\pi\sigma^2(x^2+y^2)}
$$
However if I directly construct a kernel using $G(x,y;\sigma)$ the output kernel looks completely different from first computing $g(x,y;\sigma)$ and then doing the FFT. What is the correct approach here? I hope the question is clear, but it is essentially how I can construct a gaussian kernel directly in frequency space so i do not have to FFT the kernel.
You can check numerically that fft(g) = G, that will be easier to analyze than the filtered result. At least you can easily plot it and identify if the coefficients are shifted, flipped, or scaled. One thing easy to do incorrectly is the determination of the frequency corresponding to each coefficient. I would strongly recommend that you first aproach the 1D version of your problem, here you have an example of 1D filtering I wrote for a related question.