Q:

Fourier transform on images with holes

This is a problem in our lab work recently

I would like to fourier transform an image with several "holes" on it.
Those holes are ellipses and of different orientations.
E.g. an imgae with holes
After fourier transform, some magnitudes would be modified.
These holes must be kept intact when inversing the fourier transform, also those modifications would be recovered later.
so simply padding with zeros is not working because
1) Modifications will "contaminate" the holes when inversing the fourier transform.
2) Padding with zeros and putting the holes content back will make a complete recovery impossible.

So, would it be possible to do fourier transform on an image with holes?
or could we map or decompose the concave image into rectangle(s)?

A:

Two answers depending on exactly what you want to do:
1- Can you find the spatial domain equivalent of your magnitude modification, and apply a 2D convolution on the image but the holes?
2- Can you follow an iterative approach, in which you replace the "holes" with an average value, do fft, apply your magnitude transformation, do ifft, replace the holes with the original content, and repeat until you get sufficient convergence? 

