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

  1. Jul 8, 2013 #1
    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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  3. Jul 9, 2013 #2

    Andy Resnick

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    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?
     
  4. Jul 9, 2013 #3
    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.
     
  5. Jul 9, 2013 #4

    Andy Resnick

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  6. Jul 9, 2013 #5
    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. :/
     
    Last edited: Jul 9, 2013
  7. Jul 10, 2013 #6

    Andy Resnick

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    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.
     
  8. Jul 12, 2013 #7
    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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  9. Jul 12, 2013 #8

    Dale

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    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).
     
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