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

[jasonpatel]

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.

 

[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?

 

[jasonpatel]

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.

 

[Andy Resnick]

Try these sites: remember that your images are not continuous, but sampled arrays- the 'discrete Fourier transform' is more applicable for you, I think...

http://homepages.inf.ed.ac.uk/rbf/HIPR2/fourier.htm
http://www.cis.rit.edu/class/simg782/lectures/lecture_12/lec782_05_12.pdf

does this help?

 

[jasonpatel]

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. :/

 

[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.

 

[jasonpatel]

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 don't understand how the width of the peak in pixels fits in? Any guidance?


and again THANKS!

 

[Dale]

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).