## 4 Discussion  
We propose herein an iterative CNN-based method for CS-MRI reconstruction. The method combines image-based CNNs and k-space corrections, with the CNNs representing a priori image space information. The proposed method performs iterative calculations in the image and k-space domains through image restoration using the CNNs and k-space correction. In this manner, the proposed method can reduce the error in k-space correction. Compared to those obtained using other techniques, the images that are reconstructed using the proposed approach demonstrate enhanced image quality with a reduction in aliasing artifacts. The above-described results demonstrate that the proposed method outperforms the U-net trained on the image space and other noniterative methods in terms of more effective suppression of aliasing artifacts. For example, Figure 4 reveals the reconstruction errors in the ventricles of the brain for a radial sampling mask of 10%, as observed using the U-net and noniterative methods. On the contrary, the proposed method can suppress these reconstruction errors. In addition, the results obtained using the proposed method were also improved compared with those of the other methods in terms of the PSNR and SSIM for all sampling rates of the Cartesian and radial sampling masks. These results indicate that the proposed method can enhance image quality and decrease acquisition time. Additionally, the results of the quantitative evaluation reveal that, in comparison with image-based CNNs and other noniterative approaches, the proposed method increases the PSNR and SSIM for Cartesian and radial sampling masks. Particularly, at radial sampling masks of 10% and 20%, the noniterative method does not improve image quality in terms of the SSIM when compared to the U-net and proposed methods. These results support the fact that the proposed method can reduce the error in k-space correction using iterative k-space correction.  
Several studies have already compared deep-learning-based methods with conventional CS-MRI algorithms. For example, [PERSON] et al. [22] reported that a three-layered CNN based on [PERSON]'s method [18] can achieve superior performance compared to CS-MRIs by utilizing sparsity in wavelet transforms [8] as well as dictionary learning [12]. Furthermore, the fastMRI project [32] also showed that the U-net performs substantially better than the total-variation-based CS-MRI [33]. Considering these reports, we believe that it is sufficient to compare the proposed method to the U-net and the noniterative methods.  
The major limitations of the present study include the optimization of CNN hyperparameters, i.e., the number of layers, filters, epochs, and batch size. In general, these parameters are empirically determined, and the objective optimization of these parameters is necessary. In addition, the experimental data used in this study correspond to real-valued MR images, which are different from the actual k-space data obtained from an actual MRI scanner (complex-valued MR images). Hence, an imaginary channel would have to be connected to the input and output CNN layers. Alternatively, complex-valued data would have to be preprocessed. In the future, we will focus on evaluations of clinical usefulness by radiologists and testing with other MR modalities, such as T1-weighted images.