I am trying to find the correct filter to convolve with an image so they have the same PSF.
I have the final PSFs of both images:
- A: PSF of the image with the wider PSF.
- B: PSf of the image with the narrower PSF.
(So the FWHM of B is smaller than A).
For the time being I am trying to do the most simple (naive) deconvolution: by simply dividing the Fourier transform of A with that of B. The reason is that the type of data that I am working with (astronomical images) will not have zeros in the fourier transform. Also I am using very high signal to noise images of stars (point sources) so the effect of noise is minimal. Later on, I will try to implement the Wiener deconvolution. But my problem seems to be more basic than that!
When I take the naive deconvolution steps, it seems the Final PSF I get is translated by half the image width as you can see below. Just a note: I have set the sums of both images to 1 before this process. Also, the images are all inverted so zero is shown with white.
This is A (the wider PSF):
This is B (the narrower PSF, same image size as A):
This is the Fourier transform of A:
This is the Fourier transform of B:
This is the pixel by pixel division of the Fourier transform of A divided by the Fourier transform of B. Following the advice in this stackexchange-Mathematics question, I have set all the pixels with a very small B value to zero (a plus-like figure in the center that is all white).
And this is the inverse Fourier transform of the division:
It seems reasonable, but it is not centered, its as if it has been translated by half the image size! I know I can multiply the Fourier domain results with constants to center it. But I don't understand why this happens. I am affraid I might have made a mistake somewhere! If I understand it, then applying that translation is easy.
I am using this exact algorithm for convolution (where the Fourier transforms are multiplied) and the inverse discrete Fourier transform result is centered. So, I know there is no error in my inverse discrete Fourier transform steps.
As a summary, my question is that why does the inverse Fourier transform of the frequency space appear to give a translated (by half the image size) spatial image in division?