# Image Quality Improvement and Memory-Saving in a Permanent-Magnet-Array-Based MRI System
## 2 Theory
### Effects on Image Reconstruction Time  
In terms of image reconstruction, the system in Equation (6) is solved using conjugate gradient method [16]. As shown in Figure 4 and presented in the previous sub-section, increasing sampling points and increasing angles have different effects on image quality. To further investigate these two approaches, their effects on the image reconstruction time are examined through checking the time it takes for iterations, and the relation between the NRMSE of the solution (i.e., the resultant image) and the number of iterations. Figure 5 shows the calculation time versus the number of iterations for the cases in Figure 4 at different \(N_{t}\)s and \(N_{\theta}\)s. As can be seen, the curves are all linear, close to one another, and have similar slopes. It is indicated that, although the sizes of the matrices are different, the time it takes for a single iteration is similar. The case that has the highest slope (\(N_{\theta}=180\), \(N_{t}=256\)) has a computational time of 0.135 s/iteration whereas the one that has the lowest slope (\(N_{\theta}=180\), \(N_{t}=128\)) has a computational time of 0.106 s/iteration. Figure 6a shows the NRMSE of the reconstructed images versus the number of iterations when sampling points increase (\(N_{t}=128\), 256, and 512, and \(N_{\theta}=90\)), corresponding to the images shown in column 1, 3, 6 in Figure 4. When the SNR is high (SNR = 100 dB), all the cases show a rapid drop of NRMSE when the number of iterations increases. At \(N_{t}=128\), 256, and 512, the NRMSE drops to 0.0315, 0.0302, and 0.0299 by 71.6%, 66.8%, and 85.2% after 13 iterations compared to those of the first iteration (0.1110, 0.0911, and 0.2017), respectively. After 13 iterations, the decrease rates of all the three curves are slowed down. With the same iterations, i.e., approximately the same amount of time,  
Figure 5: The relationship between time and iterations in different systems. the NRMSE in the case when \(N_{t}=512\) decreases the fastest, changing from the highest to the lowest among the three, which indicates the highest effectiveness in terms of obtaining high image quality with the same amount of calculation time. Equivalently, it means that when \(N_{t}\) is high, lower number of iterations, thus less computational time, leads to the same image quality. When SNR drops to 20 dB, rather than a rapid decreases and a knee-shape as those in the cases at 100 dB, each curve at this SNR shows a minimum and an increase to a saturation when the number of iterations increases. As shown, the NRMSEs at the first iteration are 0.1055, 0.1080, and 0.2070 at \(N_{t}=128\), 256, and 512, respectively, which are close to the values of systems with the same \(N_{t}\) when \(\text{SNR}=100\) dB. At \(N_{t}=128\), unlike at 100 dB that the NRMSE decreases rapidly, the NRMSE increases when the iteration number increases. This is due to the nature of iterative method where iterations empower noise to corrupt the result [22]. When \(N_{t}\) increases, at \(N_{t}=256\) and 512, the curves show NRMSE minima of 0.0848 and 0.1108 at 3 and 9 iterations which are a decrease by 21.5% and 46.5% compared to those of the first iteration, respectively. A higher \(N_{t}\) leads to more iterations for the system to reach a NRMSE minimum, which may be due to the fact that an increase in \(N_{t}\) increases the size of the matrix but does not increase the information to the system proportionally. Based on the observation above, an increase in \(N_{t}\) does not help to reduce the computational time for image reconstruction. For the cases when \(N_{\theta}\) increases (corresponding to the systems in column 1, 2, 4 in Figure 4, \(N_{t}=128\), and \(N_{\theta}=90\), 180, and 360), Figure 6b shows the relationship between the number of iterations and the resultant NRMSER.