I understand why we shift the Fourier transform such that the 0-frequency is centered for visualization. In the shifted DFT(u,v) of an M*N 2-dimensional image,
- the top-left corner of the 4th quadrant is (0,0) frequency or (low u, low v)
- the bottom-left corner of the 1st quadrant, (M-1,0) or (high u, low v);
- the bottom-right corner of the 2nd quadrant, (M-1,N-1) or (high u, high v); and
- the top-right corner of the 3rd quadrant, (0,N-1) or (low u, high v).
Now, when we apply filters (centered again) on the centered DFT, aren't we changing both low and high frequencies closer/further to/from the center?
For example, the centered Gaussian high-pass filter is centered at M/2 and N/2, and supposed to attenuate only low frequencies (is it?). However, applying this filter to the shifted DFT will attenuate not only low frequencies, but also high frequencies in the 1st, 2nd, and 3rd quadrants.
I did a little experiment and can confirm this effect. Applying the centered Gaussian high-pass filter to a centered DFT was not equivalent to applying the non-centered Gaussian high-pass filter to the same non-centered DFT. I had to apply the filter three more times at each corner of the non-centered DFT to get the same result.
I couldn't find any good explanations why this (high-pass filter changing high frequencies around the center or low-pass filter changing low frequencies) is acceptable.
To simplify this question, let's take a one-dimensional image as example. The non-shifted DFT has the 0-frequency on the left edge while the M-1 frequency on the right edge. The shifted DFT has the 0-frequency at the center and the M-1 frequency right next to it on the left side. Attenuating this center in effect changes both the 0 and M-1 frequencies. This is different from attenuating the 0-frequency on the non-shifted DFT.
Thanks in advance!