Bottom-line: how to create a 90 degrees out of phase signal from the real part of the fourier transform of a 1D signal (i.e., fft of a line of an image). ? *(I *(I think the Fourier shift thm can help but mot exactly sure how yet)
In MRI the actual FID echo signal is acquired in quadrature (cf. http://mriquestions.com/real-v-imaginary.html ). Also each line of the k-space (=2D Fourier transform raw data in MRI jargon) is acquired sequentially, i.e., a sampled 1D (temporal) signal (in the receiver coil). Each sample is put in each x,y position in the 2D kspace (which correspond to spatial frequencies), which is the Fourier transformed to get a spatial (visually interpretable for humans) image.
That being said, I want to take any image, for the sake of the example, lets say the artificial 'cameraman' image of Matlab. We can do a DFT of this image and get a real and imaginary image. I want to make as if/simulate that this image was MRI acquired. Therefore, as I understand the description here http://mriquestions.com/real-v-imaginary.html , one could take the real part (but it doesnt matter if it was real, it could be the imaginary, but let's take the real one for the example), the dephase it w.r.t itself by 90 degrees. Then discard the imaginary part and replace it with this "dephased" version of the real part, since the real and imag parts are supposed to be the same but just dephased by 90 degrees.
The special thing is that this will result in a "phase" image e.g.: https://www.researchgate.net/figure/a-A-256-256-MRI-head-phase-image-b-its-corresponding-residue-distribution_fig6_232321363, in the spatial domain too. I thought about the Fourier Shift theorem: https://www.dsprelated.com/freebooks/sasp/Shift_Theorem.html ... but not sure how to use it...