## 2 Proposed SALDL Based on GCM
### Proposed Layer Decomposition Learning  
In addition to the DDN, as shown in Equation (11), the residual deblurring strategy can be adopted. It is noteworthy that the DDN and residual deblurring network (RDN) were derived from the proposed GCM. In other words, both are the proposed deep-learning architectures. The RDN estimates the detail layer from two types of images, i.e., the input halftoned image and the predicted base layer via residual learning. It appears that the RDN is similar to the conventional PRL [23, 25]. However, the significant difference is that the deblurring strategy is adopted in the former. In other words, the predicted base layer is the Gaussian-filtered version of the original image, and the base layer is designed based on the GCM proposed for residual learning. This RDN can provide better performances than the DDN, owing to the effect of residual learning. However, this RDN is restricted in terms of recovering image structures clearly. Hence, a new subnetwork known as the image structure map predictor is incorporated in the proposed SALDL. Figure 5 shows the entire architecture of the proposed SALDL, which comprises two subnetworks. One is the image structure map predictor (ISMP), and the other is the SARDS. The ISMP transforms the input halftoned image into a Laplacian map, which refers to an image obtained by convolving the original image and the Laplacian filter. An example of the predicted Laplacian map is shown on the right side of Figure 5. Even though the predicted base layer can be input to the image structure map predictor, in this case, the detailed representation is not satisfactorily restored because the predicted base layer has already lost some texture information. As shown in Figure 5, the input halftoned image contains more texture information than the predicted base layer. The ISMP includes a pretrained subnetwork known as the initial reconstruction subnetwork (IRS). This subnetwork generates the initial reconstructed image from the input halftoned image. Because the input halftoned image is quantized, it is preferable to predict the image structures from the initial reconstructed image than from the halftoned image. In fact, the Laplacian map is the filtered version of the original image, which implies that the Laplacian map can be predicted by convolving the Laplacian filter with the initial reconstructed image. However, the initial reconstructed image differs from the original image; hence, more convolution and ReLU layers are required at the back of the IRS. Based on the experiments, it was confirmed that the accuracy of the Laplacian map decreased when the IRS was not adopted, rendering the predicted detail layer less accurate. Therefore, the IRS is key for increasing the accuracy of the ISMP. As shown in Figure 5, the initial reconstructed image was changed to increase the performance of the ISMP while learning the entire network. The SARDS requires three input images: the predicted base layer, the Laplacian map, and the input halftoned image. The predicted Laplacian map was stacked on the top of the input halftoned image and the predicted base layer via a concatenation layer. Subsequently, it was input into the SARDS to estimate the detail layer. \[\mathbf{x}^{(d)}=f_{\theta}^{sards}\left(\mathbf{x}^{(b)},\mathbf{x}^{(l)},\mathbf{x}^{i} \right),\;\mathbf{x}^{(l)}=f_{\theta}^{isamp}\left(\mathbf{x}^{i}\right) \tag{12}\]  
where \(f_{\theta}^{sards}\) and \(f_{\theta}^{isamp}\) denote the proposed SARDS and ISMP, respectively. \(\mathbf{x}^{(l)}\) denotes the predicted Laplacian map, and \(\mathbf{x}^{(d)}\) denotes the predicted detail layer. In Equation (12), the Laplacian map is predicted from the input halftoned image, not the base layer. Based on experiments, it was discovered that the Laplacian map was not accurately estimated because the base layer contained missing information. The use of the Laplacian map provided subnetwork \(f_{\theta}^{sards}\) with spatial information regarding areas that were flat, lined, or textured.