Magnetic resonance (MR) imaging (MRI) provides an indispensable imaging method without ionizing radiation in contemporary clinical applications. However, the scanning speed of MR imaging technique is limited. Both compressed sensing (CS) and parallel imaging (PI) techniques have been applied to reduce the MRI scanning time.
A variety of k-space undersampling patterns has been applied to reduce the amount of collected data, such as one-dimensional (1D) uniform undersampling (1DUU), 1D Gaussian random undersampling (1DGU), two-dimensional (2D) Poisson-disc undersampling (2DPU), 2D Gaussian random undersampling (2DGU), and radial undersampling. According to the CS theory [1, 17], it is feasible to reconstruct the MR images from highly undersampled measurements , since the MR images exhibit a sparsity in the wavelet transform domain and spatial finite differences. CS-MRI methods [17, 34] have been proposed to solve reconstruction problems with regularization terms of the total variation (TV) and L1 norm of wavelet coefficients. Qu et al. developed the patch-based directional wavelet (PBDW and PBDWS) [24, 21] and graph-based redundant wavelet transform (GBRWT)  to improve the reconstruction performance. In addition to fixed sparse transform methods, adaptive sparse representation-based reconstruction methods have been proposed, such as dictionary learning-based MRI (DLMRI)  and transform learning-based MRI (TLMRI) [26, 32, 36, 27] algorithms. These algorithms have been proven to attain a high reconstruction performance.
Certain methods have been proposed to exploit the nonlocal self-similarity (NSS) of image patches to improve the image quality, such as the nonlocal means (NL-means) , block-matching 3D denoising (BM3D) [3, 9], patch-based nonlocal operator (PANO) , and nonlocal low-rank (NLR)-CS  methods. The BM3D method exploits the NSS of image patches via their grouping for image denoising purposes [3, 9], which has been applied in MRI reconstruction. Qu et al.  exploited the NSS of image patches and established a PANO to reduce the reconstruction error. Dong et al.  developed the very promising NLR-CS model employing NLR regularization of similar image patches constructed though block matching (BM).
Parallel MRI (PMRI) is also a well-known technique to accelerate MRI, which is often been combined with the CS theory to improve the reconstruction performance. Sensitivity encoding (SENSE) 
explicitly utilizes sensitivity information. However, the performance of these methods is limited due to the difficulty in the accurate measurement of sensitivity information in practical applications. An iterative self-consistent parallel imaging reconstruction using eigenvector maps (ESPIRiT) model has been proposed
. In contrast to the SENSE model, ESPIRiT model estimates multiple sets of coil sensitivity. We introduced the Lp pseudo-norm joint TV regularization term into the ESPIRiT scheme to improve the reconstruction performance.
Another type of PMRI reconstruction method avoids the difficulty in sensitivity estimation by implicitly considering sensitivity information, such as the generalized autocalibrating partially parallel acquisitions (GRAPPA) [11, 2] and iterative self-consistent parallel imaging reconstruction (SPIRiT) [16, 18, 20, 8, 31, 7] methods, which relies on the k-space local kernel calibration. SPIRiT includes two reconstruction schemes in both the image [18, 20, 31] and k-space [16, 18, 8] domains. An L1-SPIRiT scheme was has been obtained by combining the regularization term of the joint L1 norm (JL1) in the wavelet domain with the k-space domain-based SPIRiT model and solved with the projection over convex sets (POCS) algorithm [16, 8]. Duan et al.  applied the fast iterative shrinkage thresholding algorithm (FISTA) to solve the SPIRiT PMRI reconstruction problem in the k-space domain with the joint TV (JTV) regularization term. Weller et al.  adopted the alternating direction method of multipliers (ADMM) technique to solve the SPIRiT PMRI reconstruction problem in the image domain with the JTV regularization term. Most recently, the STDLR-SPIRiT scheme has been established  by combining the SPIRiT model with the simultaneous two-directional low-rankness (STDLR) in k-space to realize improved reconstruction. In fact, STDLR-SPIRiT has exploited the local low-rank (LR) prior rather than the LR feature based on nonlocal image structures.
Recent methods have exploited the NSS of images by imposing the group sparsity or the low-rankness of nonlocal similar patches to improve the reconstruction quality. Among them, the NLR-based methods have achieved an excellent performance [5, 33]. In addition, there have been a variety of improved SPIRiT-based methods for PMRI reconstruction. However, to the best of our knowledge, no SPIRiT-based algorithm has applied low-rankness of nonlocal similar patches. In this paper, we propose an NLR-SPIRiT scheme incorporating NLR regularization into the SPIRiT model. The NLR-SPIRiT model fully utilizes both the NSS in MR images and the calibration consistency in the k-space domain to improve the reconstruction performance. By employing the Nash equilibrium (NE) formulation [4, 35], we reformulate the NLR-SPIRiT model into a two-objective optimization problem including a rank minimization problem and a least-squares (LS) problem. We adopt the weighted nuclear norm (WNN) instead of the nuclear norm (NN) as a surrogate of the rank, which yields a more efficient method to solve the rank minimization problem. The LS problem is efficiently solved with the ADMM technique. The experimental results demonstrate the superior performance of the NLR-SPIRiT model over state-of-the-art methods in terms of three objective metrics and visual comparison.
We organize the rest of this article as follows: in Section II, we review the SPIRiT model and SPIRiT-based algorithms. In Section III, we describe the NLR-SPIRiT model, which incorporates NLR regularization of similar patches into the SPIRiT model. We adopt the WNN as a surrogate of the rank and employ the NE formulation and ADMM technique to efficiently solve the NLR-SPIRiT model. Section IV presents the experimental results and analysis. Finally, we provide the conclusion of this paper in Section V.
Ii Related work
Suppose represents a multicoil image stacked in columns, denotes the undersampled k-space data of the multicoil image stacked in columns, is a undersampled operator selecting only the acquired k-space data from the entire k-space grid,
is the discrete 2D Fourier transform matrix,is the Fourier operator applied on single coil image, , , represents the undersampled encoding matrix, is a identity matrix, denotes Kronecker product, , and is the total number of coils. The undersampled k-space data of multicoil images are thus given by:
The SPIRiT calibration consistency equation in the image domain is expressed as:
where is the image domain-based SPIRiT operator, acquired from autocalibration signal (ACS) lines (as shown in Fig. 1(a)).
Murphy et al.  proposed the image domain-based L1-SPIRiT model by introducing the joint sparsity to improve the reconstruction quality. The image domain-based L1-SPIRiT minimization problem is as follows:
where is the coil-by-coil wavelet transform, and denotes joint L1 norm. A simple and efficient POCS algorithm is developed to solve problem (3).
The JL1 and JTV regularization terms are  combined with the image domain-based SPIRiT model, and the corresponding optimization problem is given by:
where denotes the JTV regularization term. The above optimization problem (4) can be solved with the ADMM technique.
Next, we introduce the k-space domain-based SPIRiT model. The SPIRiT calibration consistency equation in the k-space domain is expressed as:
where denotes the k-space data of the multicoil image , and is the k-space domain SPIRiT operator, which convolves a series of calibration kernels acquired from the ACS lines with the entire k-space.
The undersampled k-space data of multicoil images are determined as:
Vasanawala et al. proposed the k-space-based L1-SPIRT model by introducing the joint sparsity to enhance the reconstruction quality . The L1-SPIRiT model in k-space can be expressed as the following minimization problem:
Problem (7) is solved with the POCS algorithm . We also discussed the solution to problem (7), and proposed an algorithm with a higher reconstruction quality than that obtained with the POCS algorithm .
We introduce the JTV regularization term into the k-space-based SPIRiT model to increase the reconstruction quality , and the corresponding optimization problem is given by:
By combining the k-space-based SPIRiT model with the STDLR regularization term, Zhang et al. proposed the following STDLR-SPIRiT model :
where is a fatter matrix comprising stacked Hankel matrices in the horizontal direction, is a fatter matrix comprising stacked Hankel matrices in the vertical direction, is the Hadamard product, is an operator to convert into a block Hankel matrix, and and are the weights corresponding to the Fourier transform coefficients of sparse transform filters in the vertical and horizontal directions, respectively. Problem (9) can be solved with the ADMM technique.
Iii The proposed algorithm
Iii-a Problem formulation
Past works have verified that the NSS prior information facilitated sparsity enforcement, resulting in reconstruction quality enhancement. To further improve the PMRI reconstruction quality, we incorporate the NLR regularization term into the SPIRiT model to fully utilize the NSS of PMRI.
Suppose denotes the coil image of , and a single coil image is divided into overlapping patches of size . In each reference patch, we search for similar patches within a local window (e.g., 4040), and choose similar patches closest to the reference patch in terms of the Euclidean distance. The selected similar patches are vectorized and grouped to construct a similar patch group matrix , where is a BM operator. The step of obtaining NSS prior information is depicted in Fig. 1(c). Therefore, the PMRI reconstruction problem based on the NLR regularization term and the SPIRiT model can be formulated as the following problem:
where denotes the rank of the matrix , and are tuning parameters, which are applied to balance the data fidelity, the calibration consistency, and the NLR regularization term.
Iii-B Problem solution
Problem (10) is a large-scale nonconvex optimization problem that is difficult to solve. Authors [4, 35] have proposed employing an NE formulation. With the application of the NE formulation [4, 35], we convert the reconstruction (10) into a two-objective optimization problem:
where denotes LR approximation of the patch group matrix , and is determined as follows:
where , the adjoint operator of , places back the denoised patches at their original positions (as shown in Fig. 1(c)). is a diagonal matrix with each diagonal element equal to the time of the corresponding pixel belonging to the overlapping patches throughout .
Iii-B1 Minimization with respect to
Generally, the rank penalty objective optimization problem of
is a nondeterministic polynomial time (NP)-hard problem. Thus, given that the WNN[12, 15] may yield a better rank approximation than the NN, we adopt the WNN as a convex surrogate of the rank. The WNN of can be written as:
With the use of the WNN as a surrogate of the rank, let , problem (11) can be rewritten as follows:
be the full singular value decomposition (SVD) of, , is the singular value of , and . Hence, the optimal solution to (15) is , , where can be calculated as:
where is the soft threshold operator, . And the weight can be calculated as:
where is a constant, is to avoid dividing by zero, and then the initial can be estimated as:
Iii-B2 Image reconstruction
After calculating , the whole image can be reconstructed by solving problem (12). By introducing an auxiliary variable and corresponding Lagrange multiplier , the LS problem (12) can be converted to the following subproblems via the ADMM technique:
The solution to subproblem (19) with respect to is given by:
Let , and can thus be obtained via the direct inversion of each block in matrix , as shown in Fig. 1(b). Then, we have the update of as . is a diagonal matrix, and subproblem (20) with respect to yields the following closed-form solution:
Finally, we obtain the NLR regularization-based SPIRiT (NLR-SPIRiT) PMRI reconstruction algorithm, as expressed in Algorithm 1. And the schematic illustration of NLR-SPIRiT is described in Fig. 1. Algorithm 1 is terminated when the relative error (RE) falls below the tolerance . In Algorithm 1, the BM step is performed every iterations to reduce the computational complexity, as shown in step 4.
The optimization problem (10) can also be solved by using the variable splitting and ADMM techniques. The corresponding algorithm is called ADMM-NLR-SPIRiT. Please refer to the appendix for details.
Iv Experimental results
Iv-a Experimental setup
In our experiments, we compared the proposed NLR-SPIRiT model to state-of-the-art algorithms to solve the PMRI reconstruction problems such as: JTV-SPIRiT , LPJTV-ESPIRiT , STDLR-SPIRiT , and NLR-SPIRiT-baseline (a variant of the NLR-SPIRiT model with the standard NN). All the considered algorithms were implemented in MATLAB.
To validate the performances of all the considered algorithms, we conduct experiments on four fully sampled in vivo human datasets. The first fully sampled brain dataset  (dataset 1, as shown in Fig. 3LABEL:sub@fig3a) was acquired on a clinical MR scanner Discovery MR750 using a 12-channel head-neck spine coil and a 3D T1-weighted gradient echo sequence (matrix size = , TR/TE = 7.4/3.1 ms, field of view (FOV) = mm, slice thickness = 1 mm). We chose a single slice of each multislice dataset in our experiment. The second fully sampled knee dataset  (dataset 2, as shown in Fig. 4LABEL:sub@fig4a) was acquired on a GE scanner using an 8-channel HD knee coil and a 3D FSE CUBE sequence (matrix size = , TR/TE = 1550/25 ms, FOV = mm, slice thickness = 0.6 mm). We chose a single slice of each multislice dataset in our experiment. The third fully sampled brain dataset  (dataset 3, as shown in Fig. 5LABEL:sub@fig5a) was acquired on a 3T SIEMENS Trio system and an 8-channel head array coil (matrix size = , TR/TE = 2530/3.45 ms, slice thickness = 1.33 mm, FOV = mm). The fourth fully sampled brain dataset  (dataset 4, as shown in Fig. 6LABEL:sub@fig6a) was acquired on a 3T SIEMENS Trio whole-body scanner using a 32-coil head coil and the 2D T2-weighted turbo spin echo sequence (matrix size = , TR/TE = 6100/99 ms, FOV = mm, slice thickness = 3 mm). The 32-channel k-space data was compressed to four virtual coils.
To validate the considered algorithms, fully sampled datasets were subjected to retrospective undersampling with the above different undersampling patterns and acceleration factors (AFs) (including ACS lines). In our experiments, we employed two undersampling patterns, 2DPU and 1DGU. It should be noted that the 2DPU patterns always include 2424 ACS lines, and the 1DGU pattern includes 20 ACS lines. All experiments were simulated on a workstation machine equipped with an Intel Core i9-10900X @ 3.7 GHz processor, 64 G of RAM memory and a Windows 10 operating system (64 bit).
Three commonly adopted objective metrics were employed to evaluate the quality of the reconstructed images, namely, the signal-to-noise ratio (SNR), high-frequency error norm (HFEN) , and structural similarity index measure (SSIM) . It should be noted that the calculation of all metrics is limited to the region of interest (ROI). High SNR and SSIM values or low HFEN values indicate more accurate reconstruction. In regard to reference image and the reconstructed image , the SNR is defined as:
where denotes the mean square error between and , and
is the variance in.
The HFEN is expressed as:
where is a Laplacian Gaussian filter operator used to determine the image edges.
The SSIM is calculated as:
where and are the means of and , respectively, and are the variances of and , respectively, represents the covariance of and , and are constant.
Iv-B Parameter settings
A kernel size of was adopted in the SPIRiT-based algorithms in the following experiments. The parameters of JTV-SPIRiT, LPJTV-ESPIRiT, and STDLR-SPIRiT were manually tuned for SNR optimal. In regard to NLR-SPIRiT-baseline and NLR-SPIRiT, image patches were used, and the total number of similar patches was 43 . A reference image patch was extracted every 5 pixels along the horizontal and vertical directions to reduce the computational complexity. The main parameters were defined as follows: , , and . These settings remained fixed in all experiments.
In addition, to obtain a better parallel image reconstruction performance, the parameter and were adjusted to optimize the SNR and HFEN values. We run the proposed algorithm on the same undersampled data considering the ranges of and . We chose and based on the highest SNR and smallest HFEN values. For example, at and , these two optimal indicators (SNR and HFEN) of dataset 1 were roughly optimized.
|Algorithms||Metrics||Acceleration Factor (AF)|
|Algorithms||Metrics||Acceleration Factor (AF)|
|Algorithms||Metrics||Acceleration Factor (AF)|
Iv-C Convergence analysis
To reflect the convergence of the proposed NLR-SPIRiT algorithm, Fig. 2 shows SNR, HFEN, and RE versus the iteration number during the reconstruction of dataset 1 based on the 2DPU pattern . We observe that NLR-SPIRiT approximately reaches the maximum SNR and the minimum HFEN when RE falls below 1e-4. At this time, NLR-SPIRiT converges and then is terminated when RE falls below 1e-4.
Iv-D Comparison to previous works
Iv-D1 Comparison results of 2DPU patterns
To facilitate the assessment of the image quality (commonly a subjective parameter), Figs. 3-5 show the images reconstructed via all considered algorithms based on the 2DPU patterns , and their corresponding error maps. Fig. 3 shows that the reconstructed images via JTV-SPIRiT obvious artifacts. LPJTV-ESPIRiT, STDLR-SPIRiT and NLR-SPIRiT-baseline mitigate these artifacts to a certain extent. The proposed NLR-SPIRiT algorithm effectively removes image artifacts and preserves the details. Hence, NLR-SPIRiT achieves the best visual quality among all competing algorithms. Fig. 4 shows that the reconstructed images via JTV-SPIRiT and NLR-SPIRiT-baseline exhibit apparent artifacts. LPJTV-ESPIRiT and STDLR-SPIRiT mitigate artifacts to a certain extent. The proposed NLR-SPIRiT algorithm further removes these artifacts and succeeds in reconstructing certain details accurately, such as knee joint, whereas artifact areas remain in images more obviously reconstructed via other considered algorithms. NLR-SPIRiT produces the visually best reconstructed image based on visual comparison. Fig. 5 shows that the reconstructed images via JTV-SPIRiT and STDLR-SPIRiT exhibit apparent artifacts. NLR-SPIRiT-baseline and STDLR-SPIRiT mitigate these artifacts effectively. NLR-SPIRiT obtain fewest errors and produces the visually best reconstructed image based on visual comparison. In summary, the proposed NLR-SPIRiT model achieves the best visual performance.
To quantitatively evaluate the reconstruction performance of all considered algorithms based on 2DPU patterns with different AF values, comparison results of the SNR, HFEN, and SSIM values of the reconstructed images via the considered algorithms are summarized in Tables I-III. The best metric in each case is marked in bold.
As indicated in Tables I-III, one can see that the proposed NLR-SPIRiT method produces the best metrics for all datasets and AFs. For dataset 1, NLR-SPIRiT achieves SNR improvements of 1.34, 1.41, 1.39, and 0.71 dB on average over JTV-SPIRiT, LPJTV-ESPIRiT, STDLR-SPIRiT, and NLR-SPIRiT-baseline, respectively. In regard to dataset 2, NLR-SPIRiT achieves SNR improvements of 1.50, 1.01, 1.42, and 0.68 dB, respectively, on average. Regarding dataset 3, NLR-SPIRiT achieves SNR improvements of 1.40, 0.93, 1.09, and 0.48 dB, respectively, on average.
NLR-SPIRiT-baseline attains a better reconstruction performance than that of JTV-SPIRiT, LPJTV-ESPIRiT, and STDLR-SPIRiT except for a case (dataset 2 with ). In other words, the introduction of the NLR regularization term contributes to reconstruction performance improvement of these SPIRiT-based algorithms. Furthermore, NLR-SPIRiT improve the reconstruction performance than NLR-SPIRiT-baseline. NLR-SPIRiT achieves a significant improvement over JTV-SPIRiT, LPJTV-ESPIRiT, and STDLR-SPIRiT in terms of the three metrics.
Iv-D2 Comparison results of 1DGU patterns
We choose the same data (dataset 4) and sampling pattern (1DGU pattern at a sampling rate of 0.34, which derived from Fig. 7(f) in 22) for comparison with the STDLR-SPIRiT  algorithm. The comparison of the PMRI reconstruction performance for dataset 4 under the 1DGU pattern is shown in Fig. 6LABEL:sub@fig6f.
As shown in Fig. 6, all the considered algorithms can produce a good reconstruction. And NLR-SPIRiT achieves the least artifacts inside the image and the least errors at the edge. It is shown that the proposed NLR-SPIRiT algorithm yields the highest visual quality among all considered algorithms.
Iv-D3 Statistical analysis
We also provide violin plots for the relative metric differences between NLR-SPIRiT and the compared algorithms in Table I-IV. As shown in Fig. 7, NLR-SPIRiT significantly outperforms JTV-SPIRiT, LPJTV-ESPIRiT, and STDLR-SPIRiT in terms of SNR, HFEN, and SSIM. In addition, we can also see that NLR-SPIRiT has slightly better performance than NLR-SPIRiT-baseline.
Iv-D4 Comparison of runtime and memory demand
Next, we compare the runtime and memory demands of JTV-SPIRiT, LPJTV-ESPIRiT, STDLR-SPIRiT, and NLR-SPIRiT. Table V summarizes the runtime and memory demands of the considered algorithms in PMRI reconstruction for dataset 1 based on the 2DPU pattern (). As indicated in Table V, JTV-SPIRiT has advantages in runtime and memory demands. Compared to JTV-SPIRiT, LPJTV-ESPIRiT consumes fewer memory but requires more runtime. STDLR-SPIRiT is the most advanced SPIRiT-based method at hand, which has the highest computational complexity, requires the longest runtime, and the maximum memory. Furthermore, NLR-SPIRiT requires 0.49 GB of RAM memory for reconstruction, only approximately one-92th of the memory demand of STDLR-SPIRiT. In other words, NLR-SPIRiT outperforms STDLR-SPIRiT in terms of memory demand and runtime.
The runtime of NLR-SPIRiT is , where is the outer iteration number, is the runtime of the block-matching operation performed every iterations, is the runtime of one LR approximation, is the runtime to update , and once, and is the runtime of algorithm initialization and preprocessing. The very high computational complexity of the LR approximation and block-matching steps result in large and values. Therefore, we will optimize the LR approximation step and block-matching steps with a graphics processing unit (GPU) to efficiently accelerate the proposed NLR-SPIRiT algorithm in the future.
Iv-E Proposed NE formulation versus ADMM formulation
According to the derivations of the NLR-SPIRiT model and ADMM-NLR-SPIRiT algorithm (see the appendix), we observe that these two algorithms attain almost the same calculation time for one-iteration.
We compared the reconstruction results obtained with NLR-SPIRiT and ADMM-NLR-SPIRiT for dataset 1 based on the 2DPU pattern . As shown in Fig. 8, NLR-SPIRiT attains faster convergence and obtains a slightly higher SNR value than ADMM-NLR-SPIRiT. And we determine that they require almost the same reconstruction time every iteration. In addition, NLR-SPIRiT contains two fewer parameters than does ADMM-NLR-SPIRiT, and therefore it is easier to tune the parameters of NLR-SPIRiT. In summary, it is concluded that the proposed NLR-SPIRiT algorithm exhibits obvious advantages in practical applications.
Iv-F Further discussion about parameter settings
We found that there exist similar parameters for reconstructing the MR images of same kind, so we also offer a parameter selection method for practical application. First, we train the parameters from 15 retrospective undersampling data sets with the 2DPU pattern (AF = 5). We ran NLR-SPIRiT on the training undersampling data sets on the ranges of and , and obtained SNRs for different parameter settings. The average SNR relative difference between the maximum SNR and the SNR for each parameter setting is calculated. As shown in Fig. 9(a), the average SNR relative differences are minimized at and . Second, we ran NLR-SPIRiT on the 32 retrospective validation undersampling data sets on the ranges of and , and obtained the average SNR relative differences for each parameter setting. As shown in Fig. 9(b), the average SNR relative differences are also minimized around and . Therefore, for this kind of data sets, the optimal parameter setting was determined to be and .
In this paper, we propose the NLR-SPIRiT model, which incorporates the NLR regularization into the SPIRiT model. The NLR-SPIRiT model fully utilizes both the NSS in MR images and the calibration consistency in the k-space domain. We adopt the WNN instead of the NN as a surrogate of the rank, and employ the NE formulation and the ADMM technique to efficiently solve the NLR-SPIRiT model. Experimental results considering four vivo datasets indicate that the proposed NLR-SPIRiT algorithm almost achieves a superior performance in terms of three objective metrics and visual perception over state-of-the-art methods. In addition, we propose a parameter setting method for practical application, which selects a set of near-optimal parameters for the same kind of MR images. We will optimize the most time-consuming BM and LR approximation steps with a GPU to efficiently accelerate the NLR-SPIRiT algorithm in the future. The proposed NLR-SPIRiT algorithm is very promising in regard to PMRI applications.
The authors would like to acknowledge Michael Lustig, Daniel S. Weller, Xinlin Zhang, Kevin Epperson, Fa-Hsuan Lin, Robreto Souza, Saiprasad Ravishankar, Bihan Wen, and Weisheng Dong, for making their code or in vivo data sets publicly available. The authors would like to thank the anonymous reviewers for their comments and suggestions.
Appendix A Alternative algorithm
The PMRI reconstruction problem based on the NLR regularization and the SPIRiT model can be rewritten as follows:
Problem (28) can be solved with the variable splitting (VS) and ADMM techniques. First of all, by introducing auxiliary variables , , and , in addition to corresponding Lagrange multipliers , , , and , respectively, problem (28) is decomposed into the following subproblems via the ADMM method:
As mentioned in Section III of the manuscript, the subproblems (29) and (30) are efficiently solved with respect to and , respectively. Subproblem (31) with respect to yields the following closed-form solution:
Subproblem (32) with respect to generates a closed-form solution, which is expressed as follows:
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