FFT or iFFT

[mzaiss]

In MRzero we use the iFFT for the transform from kspace to image space. Which is from freq space to position space.
I think for the FID to NMR-spectrum you need the FFT as you go from time space to frequency space.

If you check the FID signal for a positive dB0 it looks like cos and sin with a positive w argument. Thus the positive shift with FFT makes sense.

For negative dB0 the FID signal changes and the sin has now the negative sign of the w argument.

[m-a-x-i-m-z]

Hi Moritz,

could you please explain what is bB0, is this a change of the magnetic field magnitude seen by the spins?

Also, I'm not sure I completely follow your arguments when comparing the time domain signals to the frequency spectrum to k-space and image space. As I see it, the time domain signals (FID or an echo) in spectroscopy correspond 1:1 to the k-space signals in imaging. If we postulate that the spectrum of the time domain signal can be recovered by a direct Fourier transform, then the same must be true if the signal is acquired in presence of a gradient (e.g. in a simple gradient echo experiment). The only difference that in a gradient echo experiment the spectral spreading is caused by the applied gradient rather than by chemical shifts of different chemical groups. At this point we assume that the gradient is positive if the resonance frequencies on the positive axis are increased.

So, based on this logic, the image is an FFT of the acquired k-space signals. And contrary, if one wants to "simulate" the k-space of an object (I know this isn't a proper simulation), one would need to apply an inverse Fourier transform. So we see, strictly speaking MR k-space is not a frequency space of the object but rather a conjugate of it.

Of course this critically depends on the definition of the gradient polarity. One could invert the gradient definition such that the simulated k-space would correspond to the frequency space of the object. The tricky part is actually that one of these two alternative gradient-defined vector spaces will be left-handed, that is normal rotation matrices will not work there. The latter is important for slice / FOV positioning and needs to be carefully checked on many different levels.

For now I am quite confident (but of course you never know...) that the Siemens interpreter uses a consistent coordinate system and relies on normal rotation matrices for slice positioning. And, it looks like the image reconstruction code of Frank that relies on iFFT and produces reasonable images for simulated data gives me inverted coordinates for the measured data.

--Maxim

[mzaiss]

Your reasoning makes perfect sense, just adding a gradient should not change the FFT / iFFT.

we checke pur B0 and it is really changing teh precession frequency. so all fine here.

but we could find a sign in our gradient defienition!

will be fixed soon, than consitently it is the FFT.

thanks for the discussion.