# Image Quality Improvement and Memory-Saving in a Permanent-Magnet-Array-Based MRI System
## 1 Introduction
In a PMA-based MRI system with a non-linear SEM which may have an irregular field pattern, similar to a PatLoc MRI system with encoding using non-linear SEMs, image reconstruction is done by solving a linear equation which consists of an encoding matrix and an array of acquired signals in time domain [2; 3]. In [14], the relation of the non-linearity of the SEM on the quality of the reconstructed images in a PatLoc system is documented. A similar relation in a PMA-based system is shown recently in [2]. In both [2; 14], a non-linear SEM causes different k-space from one region to another in the FoV, i.e., different signal point population. This is due to different gradient distributions in different regions. The location-dependent k-spaces are the so-called local k-spaces. An even signal point population is desired in each local k-space which leads to a clear image with less information loss. In terms of the computer memory consumption for image reconstruction, controlling the amount so that it can be handled by a handheld device (e.g., a laptop) is critical to a PMA-based system when it is targeted to in-time and on-site imaging. In a PatLoc MRI system, conjugate gradient methods have been applied, which is converges quickly and stable [14; 15; 16]. The size of the encoding matrix is determined by both the spatial resolution of the image and the encoding process, e.g., the number of sampling points. When the encoding matrix is large, it is computationally intensive. To reduce the memory consumption, solving the system in frequency domain which has been widely used for image reconstruction in a conventional MRI system [15; 16; 17], has been shown to be an effective approach as the encoding matrix becomes sparse in the frequency domain [17; 18]. In a PMA-based MRI system, the static magnetic field is supplied by a permanent magnet array. Therefore, for the SEM, rather than electrically switched/rotated in a PatLoc system, it is mechanically moved with respect to the FoV to encode the signal for imaging [2]. When the magnet array is cylindrical, it is rotated for imaging [3; 4; 5; 6; 12]. When a non-linear SEM is rotated for imaging, increasing the sampling points or increasing rotation angles for example helps to populate the signal points in local k-spaces to improve the quality of imaging. However, this process results in a dense encoding matrix in time domain, leading to a computation-intensive and time-consuming image reconstruction compared to a conventional k-space sampling and Fourier image reconstruction. Consequently, high-performance computer which is usually stationary may be needed which prevents such an MRI system from being portable. In order to improve the quality of image reconstruction in a PMA-based system where the SEM in non-linear, in this paper, the approaches of increasing the average signal point population of all the local k-spaces by increasing the sampling point or rotation angles are investigated in terms of their impacts on image quality improvement, acquisition time, image reconstruction time, and memory consumption. By taking these approaches, however, it results in an increase in the size of the encoding matrix in time domain which is already large before these approaches, leading to a significant increase in the memory required for image reconstruction. To lower the memory consumption, the transformation of the large and dense matrix to the Fourier domain (frequency domain) is proposed to apply which results in a sparse matrix. Moreover, to further reduce the memory consumption, a row-wise truncation to the encoding matrix in the frequency domain is applied. This memory-saving method will be detailed, and its effectiveness to achieve an improvement of image quality and memory saving simultaneously will be shown using numerical results. This study helps to lower the memory for image reconstruction in a PMA-based MRI system with improvement in image quality.