## 2 Materials and Methods
### MSDEPC Model Based on Deep Learning for SISR in Ultrasound Imaging  
Figure 1 shows a simplified illustration of the MSDEPC architecture for SISR and how it improves image sharpness. The LR image and its phase congruency (PC) image of the high-frequency component were used in a deep learning model. Figure 1: Schematic illustration of the multiscale deep encoder–decoder-based SISR method with phase congruency (MSDEPC) architecture. The network consists of the encoder and decoder block and total loss, which is calculated by summing the image loss and edge loss using the appropriate ratio, \(\eta\). High-frequency images contain information on areas representing edges, textures, corners, and other details, which provide information for improving the resolution when performing the SISR method. Existing edge extraction methods include Sobel, Canny, Prewitt, Scharr, Laplacian, and hybrid edge operators [29]. However, when the scale of the image changes, such as in multiresolution analysis (MRA), the position of the edge may not be consistent, or completely different problems may occur [25]. [PERSON] introduced the PC principle to identify regions where considerable changes are in phase at the point of the step in a high-variation square wave [30; 31]. [PERSON] and [PERSON] mathematically defined _PC_ by extending the Fourier series for position \(x\) as follows [31; 32]:  
\[PC(x)=\frac{W(x)\text{inf}(|E(x)-T|)}{\sum A(x)+\varepsilon}, \tag{2}\]  
where \(W(x)\) expresses a weighting function for frequency spread, \(E(x)\) is local energy, \(T\) denotes the noise compensation, \(A(x)\) is the amplitude of the Fourier component, and \(\varepsilon\) is a small constant to avoid division by zero. The PC map has been used for edge detection because it prevents robust edge details in log-Gabor multiscale analysis subsampling images [25; 30; 33]. Instead of searching for points where there is a sudden change in intensity, this approach searches for order patterns in the phase components of Fourier transforms. _PC_ defines a point as a feature in an image with a high phase order. This is similar to physiological evidence indicating that the human visual system responds strongly to points in highly aligned images with phase information. It has a series of advantages over other image feature detectors. Since the _PC_ is proportional to the local energy of the signal, it can be calculated through the convolution of the original image with a spherical spatial filter bank such as Gabor filter [33]. The accurate extraction of local structures is possible, and edge detection without distortion is also possible in each sub-band in MRA. Considering the unique characteristics of edge detection in MRA, they improved the SISR by extracting _PC_ edge maps and leveraging them to oversee the predictive accuracy during training. After combining the LR image with the _PC_ edge map, prediction operations were performed in the multiscale network. This network consists of encoder and decoder blocks, with each basic block consisting of four layers. The entire network consists of three scale encoder-decoder symmetric networks using blocks of 1\(\times\) (4 layers), 2\(\times\) (8 layers), and 3\(\times\) (12 layers). This block sequentially cascades the convolution and deconvolution layers of different lengths, and a multiscale deep encoder-decoder is constructed. Here, the convolution and deconvolution layers have the following properties: a kernel size of 3 \(\times\) 3, stride = 1, and number of filters = 32 for 4 and 8 layers and 64 for 12 layers. Each block is connected by side outputs, as in existing studies [34; 35]. The hidden layers were created based on the initial convolutional layer, batch normalization (BN) layer [36], and PReLU [37] followed into this architecture.