# Image Quality Improvement and Memory-Saving in a Permanent-Magnet-Array-Based MRI System
## 2 Theory  
### Signal Equation & Iterative Matrix Method for Image Reconstruction in a PMA-Based MRI System  
In a PMB (permanent-magnet-based) MRI system, the SEM is the field that is generated by the magnet array and is used to encode the phase of the signal. When a cylindrical magnet array (e.g., an irregular inward-outward (IO) ring-pair aggregate [6] as shown in Figure 1a,b) is used, the signal equation can be expressed in Equation (1) as follows,  
\[s_{\alpha}(t,\theta)=\int_{V}m(\mathbf{r})c_{\alpha}(\mathbf{r})e^{-i\tau\int _{0}^{t}B_{SEM}^{\mu}(\mathbf{r}\theta)d\mathbf{r}}d\mathbf{r} \tag{1}\]  
where \(t\) is time, \(\theta\) is the angle in the \(\rho-\)direction on the transversal plane that defines the rotation of the cylindrical magnet array (as shown in Figure 1a,b), \(m\) is the magnetization in an object under scan, the bold letter \(\mathbf{r}\) is the position vector, \(c_{\alpha}\) is the sensitivity map of the \(\alpha^{th}\) receive coil, \(\gamma\) is the gyromagnetic ratio, and \(B_{SEM}^{\mu}\) is the SEM generated by a magnet array at angle \(\theta\). As permanent magnets are always on, \(B_{SEM}^{\mu}\) inherently cannot be switched off. Figure 1c shows the pulse sequence in a PMB MRI system with a rotating cylindrical magnet array where each horizontal line represents a continuously-on SEM at angle \(\theta_{n}\). In a PMB MRI system, there is less control on the SEM in time domain for image reconstruction compared to those in a PatLoc system where the SEMs are generated using coils [13; 19; 20]. Moreover, the pattern of the SEM is less perfect due to the imperfection of fabrication of an magnet array or/and a variation of the quality of magnets that are used to build the array. For this reason, a PMA-based system with a non-linear field is hard to be transformed to one with a linear system by mathematical tools so iterative matrix methods in time domain [3; 18] are widely used in this situation. During image reconstruction, the FoV and time are discretized by pixels. Following Equation (1), the \(n_{\mathrm{t}}^{\mathrm{th}}\) sampling point of a signal at the \(n_{\theta}^{\mathrm{th}}\) angle can be expressed using a summation as follows,  
\[s_{\alpha}(t,\theta)=s_{\alpha}(n_{t},n_{\theta})=\sum_{n_{r}=1}^{N_{r}N_{r}}m (\mathbf{r}(n_{r}))c_{\alpha}(\mathbf{r}(n_{r}))e^{-i\gamma\beta_{SEM}^{\mu} (\mathbf{r}(n_{r}),(n_{\theta}\Delta\theta))(n_{t}\Delta t)} \tag{2}\]  
where \(n_{r}\) is the index of the position in the FoV, \(n_{\theta}\) is the index of the rotation angle ranging from \(1\) to \(N_{\theta}\), \(N_{\theta}\Delta\theta=\theta_{\mathrm{Max}}\) is the maximum rotation angle, \(N_{r}N_{r}\) is the total number of pixels in the FoV, \(\Delta t\) is the dwelling time, and \(\Delta\theta\) is the angular step of rotation. Therefore, the image can be reconstructed by solving the following linear system,  
\[s=\tilde{E}\,\hat{m} \tag{3}\]  
where \(s\) is a \(N_{t}N_{\alpha}N_{\theta}\) column vector (\(N_{t}\), \(N_{\alpha}\), and \(N_{\theta}\) are the number of the total sampling points, the number of receive coils, and the number of rotation angles, respectively), \(\tilde{E}\) is a \(M\times N\) encoding matrix (\(M=N_{t}N_{\alpha}N_{\theta}\), \(N=N_{r}N_{r}\)), and \(\hat{m}\) is a \(N_{r}N_{r}\) column vector for the magnetization of the object under scan. Based on Equations (2) and (3), the entry on the \(n\)th row and the \(m\)th column of \(\tilde{E}\), \(e_{\mathrm{mn}}\), can be expressed using Equation (4) below,  
\[e_{\mathrm{mn}}=c_{\alpha}(\mathbf{r}(n_{r}))e^{-i\gamma\beta_{SEM}^{\mu}( \mathbf{r}(n_{r}),(n_{\theta}\Delta\theta))(n_{t}\Delta t)} \tag{4}\]  
which corresponds to the \(\alpha\)th receive coil, the \(n_{\theta}^{\mathrm{th}}\) rotation angle, the \(n_{t}^{\mathrm{th}}\) time moment, and the \(n_{r}^{th}\) position. In Equation (4), the role of coil sensitivity (\(c_{\alpha}\)) and that of \(B_{SEM}^{\mu}\) for signal encoding can be seen, which is to encode the amplitude and the phase of the signal, respectively.