# 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  
The SEM supplied by a permanent magnet array usually has a non-linear gradient. It is called non-linear SEM. When it is used to encode the signal in an MRI system, due to the non-linearity, a traditional k-space for the whole FoV, called a global k-space, does not provide the flexibility to interpret the system properties of regions with high-order spatial variations. Nevertheless, it has been shown in [2] that local k-spaces which are k-spaces of sub-FoVs[14; 15] can be used to effectively analyze the region-dependent imaging properties of this kinds of non-linear SEMs supplied by permanent magnet arrays. Since both a global and a local k-space show distributions of acquired signals for imaging, they are called signal spaces in general in this paper. Figure 2c,c1,c2 show the global and local k-spaces of a 2D digital phantom (with a size of \(\text{FoV}_{x}\times\text{FoV}_{y}\)) in Figure 2a when it is encoded by the SEMs in Figure 2b,b1 with rotations. The phantom is discretized into \(21\times 21\) grids with a grid size of \(\Delta x\,\times\,\Delta y\) that are labelled in Figure 2a1 (which is one of the sub-divided FoV). The linear and non-linear SEMs that have a linear and a non-linear gradient are shown in Figure 2b,b1, respectively. They have a \(\Delta\mathbf{B_{0}}\) of \(2\,\text{mT}\) and an average field of \(100\,\text{mT}\). For the non-linear SEM shown in Figure 2b1, it is supplied by a shimmed sparse IO ring-pair magnet array [21]. For an illustration, the imaging region in Figure 2a is divided into \(3\times 3\) sub-FoV's for the generation of local k-spaces, and Figure 2a1 shows the sub-FoV's at Row 2, Column 1. When Figure 2b is rotated at a step size of \(\Delta\theta\) and with a total step number of \(N_{\theta}\), together with a total readout time of \(\tau_{i}\) and a dwelling time of \(\Delta t\), Figure 2c,c1, respectively, show the corresponding global k-space and the local k-spaces located at a \(3\times 3\) grid that corresponds to the \(3\times 3\) sub-FoV's. In the global k-space shown in Figure 2c, the boundary of the k-space is \(2k_{\rho-max}\times 2k_{\rho-max}\), indicated using blue dashed line box, where \(k_{\rho-max}\) is determined by the gradient of the SEM (denoted as \(G^{\mu}_{\text{SEM}}(\mathbf{r})\)) and the total readout time (\(\tau_{i}\)). \(2k_{\rho-max}\) is inversely proportional to the resolution of the target image, \(\Delta x\) and \(\Delta y\). As shown in Figure 2c, when a linear SEM is rotated to encode the signal, the global k-space (on a global k-space coordinate system, \(k_{x}k_{y}\)-plane) has a radial pattern with a step size in the \(k_{\rho}-\)direction, \(\Delta k_{\rho}\), and one in the \(k_{\theta}-\)direction, \(\Delta k_{\theta}\). For the resolution of k-space, \(\Delta k_{\rho}\) is determined by the size of the target image. \(\Delta k_{\theta}\) is determined by the step size of the rotation, \(\Delta\theta\) (Figure 1b). Here when the SEM has a linear gradient, \(G^{\mu}_{\text{SEM}}(\mathbf{r})\) is a constant, \(G^{\mu}_{\text{SEM}}(\mathbf{r})=G\). For the local k-spaces when a linear SEM is used for encoding, as shown in Figure 2c1, although they correspond to the k-spaces of the sub-FoV's at different locations, they are identical and are identical to the global one in Figure 2c. This is because the gradient is uniform across the FoV as a result of a linear SEM. The insert in Figure 2c1 shows a zoomed-in view of the local k-space for the sub-FoV from Row 2, Column 1, on a local k-space coordinate system, \(k_{x}^{\text{locloc}}\)-plane. As shown, the length and direction of the k vector in local k-space, is determined by the total readout time and the gradient of the SEM, \(G_{\text{SEM}}^{\mu}(\mathbf{r})\), which is a function of location. Higher the gradient and/or longer the dwelling time, larger \(\Delta k_{\rho}^{\text{loc}}\) is and longer the distance between two successive signal points along the \(\rho\)-direction.