# ACF expression using FFT method

It is known that the ACF of a 2D digital image can be obtained by computing the inverse Fourier transform (FT) of the product of the FT of the image and its complex conjugate. That is:

My problem is that I cannot figure out the expression for the inverse of G(p,q). I’d appreciate any idea, help, or suggestion. Thank you.

• but don't you have $G(p,q)$ written ? – Ahmad Bazzi Jul 23 '19 at 7:03
• Oh no! i'm seeking the inverse of $G(p,q)$. I've edited accordingly. – oma11 Jul 23 '19 at 9:59
• but $\vert f(x,y) e^{-j 2 \pi (\frac{px}{M}+\frac{qx}{N})} \vert = \vert f(x,y) \vert$ – Ahmad Bazzi Jul 23 '19 at 10:03
• That can't be true. You're saying the absolute value of the Fourier transformed image is the same as the original image? – oma11 Jul 23 '19 at 10:21
• Given that $|e^{-j2\pi(\frac{px}{M} + \frac{qy}{N})}| = 1$, I can see what you mean. However, I can't see how that $\sum_{x=0}^{M-1}\sum_{y=0}^{N-1}{|f(x,y)|^2} = G(p,q)$. At least I've tried that in Matlab and it doesn't give me the same result. What can i be doing wrong? – oma11 Jul 23 '19 at 10:58