Fourier Transformation - image local patch

Question

Still I am not clear about the Fourier transformation. I know it represents the frequency information of the images and I can reconstruct an image using the fourier transformation.

Say, I have an image I(x,y). Its fourier transformation is F(I). I want to reconstruct a small rectangular area in that image starting from (x1,y1) and ending at (x2,y2) without reconstructing the whole image.

Is it possible to reconstruct only a small patch from F(I)?

Answer 1

To answer your question, yes it is possible; do an inverse FFT and then crop the image normally. If it seems like a cop-out it is because you're attempting to do a time-domain task in frequency domain which isn't going to be very natural.

If you insist that the calculation be done in frequency domain I think you should be able to phase shift the image to the origin (x1 + y1) then inverse FFT and discard samples outside (x2 - x1, y2 - y1).

The fundamental problem is that in frequency domain each bin (or pixel for a 2D FFT) represents a frequency and phase across the entire image in time domain. Discarding a single pixel in frequency domain results in a loss of that frequency information for the whole image and cannot be localized.