# Image Quality Improvement and Memory-Saving in a Permanent-Magnet-Array-Based MRI System
## 2 Theory
### Effects on Image Quality Improvement
Although the images at SNR = 20 dB are relatively blurry compared to those at SNR = 100 dB, when \(N_{\theta}\) increases, the reconstructed images show more details, especially for the peripheral area, and the boundary becomes clearer. It is worth noting that the noise of the image does not increase significantly with an increase in the rotation angles degrading the image quality (i.e., lowering the SSIM), unlike the cases when the number of sampling points increases. By this comparison, increasing the number of angles shows improvement on image quality for different SNRs. At a relatively high SNR, the reconstructed image becomes clearer with an increase in \(N_{t}\). For the cases at low SNRs, the improvement on image quality mainly is shown by clearer patterns in peripheral areas. Both increasing \(N_{t}\) and increasing \(N_{\theta}\) show significant improvement on image quality but the behaviour is different especially when the SNR is low. In the view of the encoding matrix of the system, \(\vec{A}\) in Equation (6), both approaches help to enlarge the size of encoding matrix where more information is contained. Therefore, a further comparison of the reconstructed images is conducted when the size of the encoding matrix is the same. The second and the third blue boxes from left to right in Figure 4 show the cases when the matrix size is 23,040 \(\times\)16,384 and 46,080 \(\times\)16,384, respectively. when the SNR is high, in each box, the system with more rotation angles always gives a better image. At SNR = 100 dB, going across the boxes when the size of the matrix increases, the average SSIM increase. However, when SNR drops to 20 dB, the situation is complicated. As show in the second box where the number  
Figure 4: The reconstructed images of brain phantom (a grid size of 0.78125 mm, a discretization of 128 \(\times\) 128, a FoV of 100 mm \(\times\) 100 mm) in numerical simulation when the non-linear SEM in Figure 2b1 was used. The SNRs of the first and the second rows correspond to the encoding at 100 dB and 20 dB, respectively. Different columns corresponds to the encoding with different numbers of rotation angles and different numbers of sampling points which is labeled on the top of first row. The systems with same size of \(\hat{E}\) are organized in the blue box. The original phantom is collected by a 3T conventional MRI system. of rows in \(\vec{A}\) is twice of that of the original system (in the first blue box), the image of the system with more sampling points (\(N_{\theta}=90\), \(N_{t}=256\)) has a high tissue contrast thus showing clearer tissue boundaries and more anatomical information. With the same matrix size and when \(N_{\theta}\) is reduced and \(N_{t}\) is increased (\(N_{\theta}=180\), \(N_{t}=128\)), although the image has a high SIMM which indicate a high image quality, it is blurry where the tissue contrast is low and the tissue boundary is not clear. When the size of encoding is increased to 46,080 \(\times 16\),384, as shown in the third blue box, the system with the maximum \(N_{\theta}\) (\(N_{\theta}=360\), \(N_{t}=128\)) gives the highest SIMM whereas the one with the maximum sampling points (\(N_{\theta}=90\), \(N_{t}=512\)) give the lowest SIMM. The later case is even worse compared to the qualities of the images when the number of rows is halved, i.e., those in the second blue box when the size of \(\vec{A}\) is 23,040 \(\times 16\),384. For the case When the system have twice of the sampling points and twice of the rotation angles (\(N_{\theta}=180\), \(N_{t}=256\)) comparing to the original system (highlighted in orange background), the reconstructed image shows an improvement in image quality with a higher SIMM (an increase from 0.412 to 0.433 by 5%) and a significant enhancement of tissue contrast with a much clearer tissue boundaries.