Using nonlinear spatial-encoding is a potential way to relax the requirements on $$$B_0$$$ homogeneity and to do magnetic resonance imaging (MRI) without gradient coils. For the reported flexible spatial-encoding strategy¹, the inverse calculation of the encoding matrix is time-consuming and there is a blurry area at the center of the reconstructed image. Here, we proposed to separate the nonlinear encoding matrix and to reduce the size of a resultant dense matrix for accelerating calculation. Furthermore, QR method is applied to eliminate the blurry area in the reconstructed image.In a low-field MRI system using permanent magnet Halbach array², the image is reconstructed based on the signal equation in (1). In our proposed algorithm, to reduce the calculation time of inversing the encoding matrix $$$E$$$, $$$E$$$ is separated into matrices $$$T\left(t,\theta\right)$$$ and $$$A\left(t,\theta,\vec{r}\right)$$$, $$$E=TA$$$ where the results of $$$A^{-1}$$$ can be reused for imaging to save calculation time and its size is reduced for further calculation acceleration. Following are the explanations. Matrix $$$T$$$ reflects the temperature influence whereas $$$A$$$ consists of$$$n_c\times{n_\theta}$$$ submatrices, $$$A_{\theta_j,c_k}$$$ as expressed in (2). Matrix $$$T$$$ is $$$m\times{m}$$$ where $$$m$$$ is the total numbers of sampling points and can be calculated using (3). In (3), $$$n_s$$$ is the number of sampling points. Matrix $$$A$$$ is $$$m\times{n}$$$ where $$$n$$$ is the total number of points in the physical domain. Therefore, for an imaging setup, matrix $$$A$$$ is fixed while $$$T$$$ changes due to the variation of temperature. So for image reconstructions, $$$A^{-1}$$$ needs to be calculated only once while the calculation of $$$T$$$ needs to be updated for each image reconstruction. Fortunately, calculating $$$T^{-1}$$$ does not consume much time or memory because $$$T$$$ is a diagonal matrix. On the other hand, $$$A$$$ is a dense matrix and calculating $$$A^{-1}$$$ is time consuming. Here, we propose to further accelerate the calculation by reducing the size of $$$A$$$ as follows. Matrix $$$A$$$ has $$$m\geq{n}$$$. To reduce the size of $$$A$$$, $$$m$$$ is reduced to approach $$$n$$$, making rank($$$A$$$) approach $$$n$$$. Based on (3), $$$m$$$ is determined by ,$$$n_\theta$$$, $$$n_c$$$, and $$$n_s$$$, and $$$n_s$$$ is determined by the sampling frequency,$$$f_s$$$. In our method, the smallest $$$m$$$ is searched for the smallest size of $$$A$$$ at different sampling frequencies when a full rank is obtained.Fig. 3 (a) shows a 2D example where eight coils are used and a chess board at the center is the object for imaging. Fig. 3 (b) shows the $$$B_0$$$ fields used for encoding. The 2D space is discretized as $$$n=52\times{52}$$$. The sample time is limited by $$${T_2}^\star$$$. When the bandwidth is set at 30 KHz, the received pulse is at the order of 30 µs. When the sample time is 32 µs, let $$$f_s$$$ = 0.125, 0.25, 0.5, and 1 MHz, and thus $$$n_s$$$ equals to 4, 8, 16, 32, respectively. Let $$$n_\theta$$$ vary from 1 to 64 ($$$\theta=0\sim{2}\pi$$$) so that $$$m$$$ varies. Fig. 3 shows the resultant rank($$$A$$$) versus $$$m$$$. As shown in Fig. 4, when $$$f_s$$$ = 0.25 MHz, $$$m$$$ = 4096 and $$$A$$$ has a full rank, which is the smallest among the cases. In Fig. 4, when $$$f_s$$$ = 0.125 MHz, more rotating angles are needed to have a full rank whereas when $$$f_s$$$ > 0.25 MHz, a larger $$$m$$$ is needed for a full rank. Based on the comparison, the size of matrix $$$A$$$ can be reduced by decreasing the sampling frequency, so is the calculation time for inversing $$$A$$$. To inverse $$$A$$$, QR decomposition³ (QR) is applied. Fig. 5 shows the image being reconstructed and the reconstructed images using QR and Kaczmarz iteration² (Kac1 and Kac2). For QR and Kac1, $$$f_s$$$ = 0.25 MHz, $$$n_\theta=62$$$, $$$n_s=8$$$ and for Kac2, $$$f_s$$$ = 0.05 MHz,$$$n_\theta=62$$$, $$$n_s=16$$$. Kac2 has longer sample time and more sampling points. As shown in Fig. 5, the imaging quality is much better in the QR-reconstruction, and most importantly, the blurry central area is significantly eliminated compared to both Kac1 and Kac2, although the reconstruction quality of Kac2 is improved by increasing the sample time and $$$n_s$$$ compared to Kac1. We successfully separate the encoding matrix for flexible spatial-encoding and reduce the size of a resultant dense matrix for calculation acceleration. The QR method is applied for image reconstructions, the blurring area at the centre of the reconstructed image is effectively eliminated, and the quality of the reconstructed image is considerably improved. The proposed approach will significantly increase the imaging capability of low-field portable MRI systems using spatial-encoding.