# Mathematical description of the ACF using fft2

I computed the acf of an image with the following code:

%# read in image

% get size of image
[N, M] = size(Img);

%# convert to double
I = double(Img);

%# subtract mean
I = I-mean(I(:));

%# normalize magnitude
I = I/sqrt(sum(I(:).^2));

%# compute acf
fft_I = fft2(I);
Acf = real(fftshift(ifft2(fft_I.*conj(fft_I))));


Now, i have to mathematically describe the procedure, and I am thinking that the FFT equation given in https://uk.mathworks.com/help/matlab/ref/fft2.html (scroll to the bottom) isn't enough?

EDIT: Please find the added mathematical description i have come up with and kindly help me verify its correctness.

• Don't you have to pad extra data to compute the auto-correlation using the FFT? – Ben Jun 19 at 15:14
• Thanks @Ben. I'm aware of that. – oma11 Jun 20 at 12:45
• i think you need to zero-pad your data to twice the length before the FFT and then the resulting ACF will have a triangular envelope inherently applied to it. McLeod and Wyvill call this "ACF type 2", – robert bristow-johnson Jun 21 at 0:04

When the signal is assumed to be ergodic, then its ACF can be computed using time averages which can also be computed using the following convolution of length $$N$$ sequence $$x[n]$$ by its conjugate symmetric version:

$$\hat{r}_{xx}[m] = \frac{1}{N} ~~~x[m] ~~\star ~~ x^*[-m] ~~$$

Taking the $$2N-1$$ point DFT of both sides yields the following:

$$\hat{R}_{xx}[k] = \frac{1}{N} ~~~X[k] X^*[k] = \frac{1}{N} |X[k]|^2$$

In a MATLAB implementation, an fftshift is required to bring $$r_{xx}[0]$$ to the center position for convenience. Hence the ACF will be obtained by an $$2N-1$$ point inverse DFT/FFT as

rxk = (1/N)*real(fftshift( ifft( abs(fft(x,2N-1)).^2 ,2N-1) ) );

The 2D case follows very similarly to this 1D outline...

• Thank you @Fat32. If i get you correctly, it means that abs(fft(x)).^2 is the same as fft(x).*conj(fft(x)) ? – oma11 Jun 20 at 12:44
• @oma11 Yes, that's correct. – Peter K. Jun 20 at 15:41
• i think it might be appropriate to caution you guys about the need to zero-pad your data to twice the length before the FFT and then the resulting ACF has a triangular envelope inherently applied to it. McLeod and Wyvill call this "ACF type 2". – robert bristow-johnson Jun 21 at 0:02
• @robertbristow-johnson I think that's not necessary (or I didin't understand your statement) as the FFT function will already be inherently doing the zero-padding when you take 2*L-1 point FFT of an L point sequence ? – Fat32 Jun 21 at 10:52
• @Fat32, the FFT doesn't zero pad. if you have $L$ samples and you're DFT is expecting $2L$ samples, somehow you gotta define all of those $2L$ samples. – robert bristow-johnson Jun 21 at 19:10