# Image Quality Improvement and Memory-Saving in a Permanent-Magnet-Array-Based MRI System
## 2 Theory
### Effects on Image Quality Improvement  
To investigate the effects on the image quality by taking the two approaches, i.e., increasing the number of sampling points (with a fixed total readout time) or increasing the number of rotation angles, numerical experiments were conducted. The non-linear SEM with monotonsity along the horizontal direction in Figure 2b1 were used for encoding the signal when a head phantom with a grid size of 0.78 mm was used. The sensitivity of receive coil was set to be uniform to exclude the encoding effect from the coil. The phantom was discretized into 128 \(\times\) 128 and the field of view (FoV) is 100 mm \(\times\) 100 mm. The SNR was set to 20 dB to mimic the situation in a low-field PMA-based system. Another case with a SNR of 100 dB was calculated for a reference. The image reconstruction in this part was conducted in the time domain where the system in Equation (6) was used. A workstation with a 256 GB RAM was used for calculation and image reconstruction. For image reconstruction, conjugate iteration method was used. Figure 4 shows the reconstructed images. The first row shows the cases when SNR = 100 dB and the number of iterations was 10 when solving Equation (6) where the structural similarity index (SSIM) reach a maximum. The second row shows those when SNR = 20 dB and the number of iteration was 4-6 when the SSIM is the highest. The normalized root-mean-square error (NRMSE) and structural similarity index (SSIM) of each reconstructed image is labeled at the bottom in red. In Figure 4, column 1, 3, 6 with a blue background show the reconstructed images when the number of sampling points (\(N_{l}\)) were set to 128, 256, and 512, respectively. The number of the rotation angles (\(N_{\theta}\)) was set to 90 and the maximum rotation angle was set to be \(2\pi\). As shown in the first row of Figure 4, at SNR = 100 dB, when \(N_{l}\) increases from 128 to 512, the SSIM d from 0.826 to 0.893 by 8.1%. When the SNR drops to 20 dB, the reconstructed images degrade dramatically. For example at SNR = 20 dB, \(N_{\theta}\) = 90, and \(N_{l}\) = 128, as shown in the sub-figure at Row 2, Column 1 in Figure 4, most of the details in the image is too blurry to be observed. Along the same row at SNR = 20 dB, when \(N_{l}\) increases, the central area of reconstructed image becomes clearer where the contrast increases and more details of the phantom become distinguishable and can be clearly seen. However, on the other hand, it is observed that the image quality in terms of SSIM is lowered from 0.412 to 0.373 by 9.47% with an increase in \(N_{l}\). This is owing to the fact that by an increase in \(N_{l}\), the noises are amplified. For this group of comparison, it can be seen that at different SNR levels, the effect and the effectiveness of improving the image quality by increasing the number of sampling points are different. At a relatively high SNR, with an increase in \(N_{t}\), the quality of the image is improved in terms of SSIM without noticeable improvement in the details of the image that can be seen. At a relatively low SNR, with an increase in \(N_{t}\), the details of the images becomes more distinguishable and clearer while the SSIM decreases due to an amplification of noise when more sampling points are added. The results when the number of rotation angles increases are shown in columns 1, 2, 4 with a yellow background in Figure 4 where the number of rotation angles were set to 90, 180 and 360 and there were 128 sampling points. As shown, at SNR = 100 dB, the SSIM of the reconstructed image is improved from 0.826 to 0.998 by 20.8% with an increase in the number of rotation angles. At the same SNR, a notable improvement of image quality occurs when \(N_{\theta}\) is increased from 90 to 180 where the SSIM of the corresponding image increases from 0.826 to 0.981 by 18.7%. This tendency is kept when the SNR decreases to 20 dB where the SSIM increases from 0.412 to 0.486 by 17.9%.