I'd like to understand the convolution theorem for Fourier transforms applied to images and kernels.
The theorem states that $F(f *g) = F(f) \cdot F(g)$, where F is the Fourier transform operator, * denotes the convolution operator and $\cdot$ elementwise multiplication. I struggle to interpret this relationship for images where I apply a convolution in form of a filter matrix, for instance like the Sobel operator.
I think in this case, f is my image and g is the filter matrix. However, I don't see, why $F(f)$ and $F(g)$ should have the same dimensionality and if they don't I cannot apply elementwise multiplication. I wondered whether I should pad the kernel with zero elements until it has the same dimensionality as the image but I'm not sure whether this is the correct way.
Can someone please help me understand this?