Suppose I have a $4x4$ image with the following values as its grey-level intensity for each pixel like this:
I want to get its Fourier spectrum. Usually, I would just punch into Matlab and run a fft for everything to be calculated automatically. Now, if I don't have Matlab, and I have to do it by hand and with just an ordinary scientific calculator, can I still do it?
I know I could do the brute force way on the Fourier transform's equation like this:
In this case, $N=M=4$.
So, $f(x,y)$ returns the intensity value of the above image at position $x$ and $y$. Eg, $f(1,1)=1$, $f(2, 0)=3$
But this will be insane to work out the summations so many times on paper. Most calculators can't do summations with complex numbers too.
I have tried doing it with Fourier transform's separability property, but doesn't seem to work as I still have to go through the tedious summations.
What other ways can I use to calculate Fourier transform on a simple image without using Matlab, particularly for the values in the first row of the example image above?