Figure 2.

Comparison between shift undersampling and static undersampling. The central panel shows the original T1-relaxation signals at the two aliasing voxels. With static undersampling (left panel), the two signals are directly combined, leading to a complete overlapping between their spectra. The associated inverse problem is severely ill-conditioned with a condition number of 3×10¹⁷. With shift undersampling (right panel), each signal is firstly multiplied by a modulating function with a unique modulation frequency, yielding a smooth signal in the first voxel and an oscillating signal in the second voxel. The two modulated signals are then combined, resulting in a spectrum with two separate spectral peaks, suggesting less spectral overlapping. The associated inverse problem is well-conditioned, with a condition number of 40.