## 2 Materials and Methods
### Gradient-Laplacian Attention Module  
[PERSON], [PERSON] and [PERSON] [20] introduced a U-shaped transformer containing Sobel pixel attention and channel attention modules for LDCT denoising. Their experiment mainly focused on the architecture of the internal structure of their dual-attention block: (i) series vs. (ii) parallel. Unlike the common multi-head self attention (MHSA) for the PViT in Figure 3a, we propose the use of a parallel dual attention block that focuses on the use of a gradient and Laplacian operation, as shown in Figure 3b.  
We compute the edge map for the input feature map \(X\) using gradient kernels, \(G_{i}\), which includes horizontal, vertical, and diagonal edge detection variation to allow detecting image edges in multiple directions. For the Laplacian operator, we use an 8-neighbor Laplacian kernel for a stronger edge detection, \(L_{8}\), instead of the typical \(3\times 3\) kernel, \(L_{3}\), to compute the second-order derivative in capturing regions with abrupt intensity changes. Equation (1) defines the computation of the edge magnitude map \(M\) using both gradient and Laplacian operators:  
\[M=\sqrt{\sum(X\times G_{i})^{2}+(X\times L_{8})^{2}} \tag{1}\]  
where \(X\) represents the input feature map that undergoes edge detection. The gradient kernels \(G_{i}\) (where i denotes the directions: horizontal, vertical, and diagonal variations) allow for detecting edges in multiple directions. The Laplacian operator is applied using an 8-neighbor Laplacian kernel (\(L_{8}\)) instead of the standard \(3\times 3\) kernel (\(L_{3}\)), providing stronger edge detection by capturing higher-order intensity variations.  
The resulting edge map \(M\) is used within the gradient-Laplacian attention module (GLAM) to refine attention weights, emphasizing the structural details in the transformer's self-attention mechanism. It is important to note that \(X\) in this equation is not the attention score matrix \(QK^{T}\); rather, it is the feature representation upon which edge detection is performed before computing attention weights.  
Hence, the overall attention module using only the proposed edge detection gradient-Laplacian kernels is as follows:  
\[Attention(Q,K,V,M)=softmax\bigg{(}\frac{QK^{T}+\beta M}{\sqrt{d_{k}}}\bigg{)}V \tag{2}\]  
where \(\beta\) balances the contribution of all edge directions in the attention modules as the model is being trained.  
Figure 4 shows how each filter highlights or detects the edges of the image edges and image texture at the marked checkpoints on the system design in Figure 2. Please note that, at these checkpoints, an N2N reshaping module is applied in order to see the actual feature map of the whole CT image and not just the patched images. The gradient operator emphasizes gradient-based edge detection by capturing directional changes in intensity--such as vertical, horizontal, and diagonal edges. This makes it particularly effective in detecting sharp transitions between regions of different densities or intensities, such as the boundaries between bones and soft tissues in CT images. The Laplacian operator, on the other hand, highlights areas of rapid intensity change by focusing on second-order derivatives, effectively enhancing fine details in the image and preserving subtle features like small structures and textures that might otherwise be lost during denoising. Further discussion about the results is provided in Section 3, including quantitative results.