# Inverse Fourier Transform Of K-space Image…what is the object space sc

Checked around a buch and could not find any help. But I needed help with:

Understanding that if I get the Inverse FT of K-space data, what is the scaling on the X-space (object space) resultant image/data i.e. for every tick on the axis, how do I know the spatial length?

More detailed explanation is attached as a image.

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Andy Resnick
Not sure if this is a homework problem.

The fourier transform pair x and ζ are related as k/z(xζ) where k is the wavevector and z the distance from source to detector; does this help?

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Wait where the relation? The equals sign?

It does help though! Can you direct me to where I can find that relation?

Or where I can find a explanation for it. Its not a homework problem (i just made the pdf to make things easier rather than try to explain everything in words); it is part of some side research and I have very little experience with fourier transforms and even less experience with experimental aspects of it.

Hmmmm, not so much. I have read quite a bit of literature but I am really perplexed because the ccd imaging the fourier plane has a spatial dimension aspect; the pixel size.

Also the frequency domain should span an infinite plane.

I am just pretty confused. :/

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Andy Resnick
Vollmerhausen and Driggers' excellent book "Analysis of Sampled Imaging Systems" may be of help to you. Sampled systems can be quite complex, since they are not linear shift-invariant systems.

While the pixel size is indeed finite, the usual interpretation is that the pixel size (say, dx) corresponds to a resolution limit in k-space (dk) and that sampling the signal can be treated as point-wise events, which is the reason for terms like x/N in DFT equations. Windowing k-space should not cause a conceptual problem.

Ok, so firstly thanks so much for your help...I will def look into that book because this is something that seems simple but has been giving me some trouble.

Secondly I have wrote down the solution (ATTACHED PDF) that one of the guys in my group gave me. But to be honest I don't understand the very first relation (in step one).

I specifically dont understand how the width of the peak in pixels fits in? Any guidance?

and again THANKS!

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Dale
Mentor
Understanding that if I get the Inverse FT of K-space data, what is the scaling on the X-space (object space) resultant image/data i.e. for every tick on the axis, how do I know the spatial length?
K-space and image space are related as follows:
BW = N Δk
FOV = N Δx
BW = 1/Δx
FOV = 1/Δk

Where N is the number of samples, Δx is the spatial tick size (i.e. spatial resolution), Δk is the k-space tick size, FOV is the total extent of the spatial image (i.e. field of view), and BW is the total extent of the k-space image (i.e. "bandwidth", but spatial frequency rather than temporal frequency).