Discrete Fourier Transform for beginners
Yes, m and n are the dimensions of the image.  F(u,v) will be a complex-valued (real and imaginary) function of u and v.
Yes your equation is correct and F(u,v) is the result of the forward DFT. It is correct that F(u,v) has real and imaginary parts, and the multiplication of that into the complex sinusoid will result in a quantity that also has real and imaginary parts.


So how do you get back a real valued image if each of the numbers in the sum over u and v is complex? The key is that F(u,v) has certain symmetry properties due to f(i,k) being real valued.  This symmetry results in the sum over u and v adding up to a real number.

The same thing happens with the one-dimensional DFT.  In that case a real valued signal f(t) produces a conjugate symmetric DFT F(u) (F(u) = F*(N-u)). The symmetry is a bit more complicated in 2D, but that is why this works out.

I often see students manipulate the DFT coefficients and accidentally destroy the conjugate symmetry. Then when they take the inverse DFT the result is complex valued when it should have been real valued.

Note that numerical errors will sometimes produce an inverse DFT that has a very small imaginary part, even though the symmetry is correctly preserved.  In these cases it is best to just take the real part as your result.