# Image Quality Improvement and Memory-Saving in a Permanent-Magnet-Array-Based MRI System
## 2 Theory
### Characteristics of a Non-Linear SEM Supplied by a PMA
In the case when the SEM is linear, as shown in the insert in Figure 2c1, the signal points along the \(\rho\)-direction is evenly distributed. For the resolution in the \(\theta\)-direction, \(\Delta k_{\theta}^{\text{loc}}\), it is determined by the rotational step of the magnet, \(\Delta\theta\), and the direction of the gradient with respect to the x-axis which is set as a reference for rotation. In the case when the SEM is linear, the direction of gradient is aligned with the x-axis, which leads to that the angle of the radial line in the local k-space, \(\theta_{ki}\), is the same as the rotation angle of the magnet. Figure 3 shows the magnitudes and angles of the gradient at the center of each FoV's when the magnet rotates from \(0^{\circ}\) to \(360^{\circ}\) for both the linear and the non-linear SEM. As can be seen in Figure 3, when the SEM is linear, the magnitude and the angle of the gradient from each sub-FoV are identical because \(G_{\text{SEM}}^{\mu}(\mathbf{r})\) is the same crossing over the FoV. This leads to an identical local k-space for each sub-FoV. Moreover, in the case when the SEM is linear, as the direction of the gradient is along the x-axis, the angles of the gradient are the same as the rotation angles of the magnet, as shown in each sub-figure in Figure 3.  
Figure 2: A 2D digital phantom and its global and local k-spaces when spatial encoding magnetic (SEM) with linear and non-linear gradients are used for signal encoding. When the non-linear SEM in Figure 2b1 is used for encoding, Figure 2c2 shows the \(3\times 3\) grid of local k-spaces. The blue dashed boxes whose sizes are determined by the linear SEM case are kept to be there as a reference. As can be seen, the k-space in each FoV is different, in terms of size, resolution, and the distribution of signal points, due to the nature of a non-linear SEM which is that the gradient is not uniform crossing the FoV, i.e., it is region dependent. This is clearly shown by Figure 3 where the magnitudes and angles of the gradients at the centers of the sub-FoVs are different from one another. When the magnitudes are large, the signal points are spread out whereas when they are small, the signal points are more concentrated within a region. Moreover, when the angle of gradient deviates away from the x-axis, the angle of the radial line in the local k-space (\(\theta_{ki}\)) becomes different from the rotation angle of the magnet array. The inserts in Figure 2c2 show zoomed-in views of the local k-spaces for the sub-FoV from Row 1, Column 1 (called k-space\({}_{11}\)) and that from Row 2, Column 2 (called k-space\({}_{22}\)). As can be seen from the first insert, the signal points in k-space\({}_{11}\) are much more spread out compared to the linear SEM case. They go outside the reference blue dashed box, which is due to the large magnitudes of the gradients at certain angles, as shown in the sub-figure at Row 1, Column 1 in Figure 3. The radial lines are not evenly distributed in the \(\theta-\)direction, which is due to the fact that the angles of the gradient deviate from the rotation angles of the magnet array. As shown in the same in sub-figure (Row 1, Column 1) in Figure 3, the deviation depends on the rotation angle. For the second insert, on the other hand, the signal points in k-space\({}_{22}\) are concentrating in a small region in the box. This is because the magnitudes of the gradients are much smaller at most of the rotation angles, which is shown in the sub-figure at Row 2, Column 2 in Figure 3. For the distribution of  
Figure 3: The magnitudes and angles of gradients at the center of each sub-FoV when the magnet rotates from 0\({}^{\circ}\) to 360\({}^{\circ}\) for both the linear and the non-linear SEM in Figure 2b,b1, respectively.