Reconstruction in the frequency domain
This type of parallel reconstruction is more complicated to understand. As in all parallel imaging techniques, the k-space is undersampled.
The missing intermediary k-space lines are calculated from the signals recorded by the different coil elements. To do so, they are combined by weighting each coil signal.
In the first generation algorithm (SMASH: SiMultaneous Acquisition of Spatial Harmonics), the weighting coefficients attributed to each element are estimated so that the combination of signals recorded by the elements, taking account of their relative location in space, simulates the effect of an intermediary phase gradient to compensate for the absence of one of the harmonics.
In the second and third generation algorithms (auto-SMASH, VD-AUTO-SMASH, GRAPPA), a part of the (missing) intermediary lines in the center of k-space is acquired, serving as a “model” to find the coefficients to attribute to the signals from each element, to reconstruct the other intermediary lines. Acquisition of these additional lines corresponds to self-calibration, the drawback being a relative lengthening of scan time, which still remains very short.