## 2 Materials and Methods
### Evaluation Metrics and Loss Functions  
Evaluating the effectiveness of LDCT denoising methods is crucial, and using metrics like Peak Signal-to-Noise Ratio (PSNR), Mean Squared Error (MSE), and Structural Similarity Index Measure (SSIM) are important for assessing both the quantitative and qualitative aspects of denoised images. Each metric provides distinct insights, which together give a more comprehensive evaluation. As for the objective functions used, using thr Mean Squared Error (MSE) and the Structural Dissimilarity Index Measure (DSSIM) as loss functions can be effective in LDCT denoising because they offer complementary advantages in training deep learning models. The Peak Signal-to-Noise Ratio (PSNR) is widely used to measure the similarity between an original high-quality image (usually the normal-dose CT) and the denoised image. The PSNR is calculated based on the ratio of the maximum pixel intensity to the noise power, measured in decibels (dB) [21], shown mathematically in Equation (3). A higher PSNR value typically indicates less noise and closer resemblance to the reference image. However, the PSNR alone does not account for perceptual quality, such as structural details and textures, which are crucial in medical imaging. Therefore, the PSNR is often used alongside other metrics. \[PSNR(x,y)=10\log_{10}\frac{(\max(x_{j}))^{2}}{MSE} \tag{3}\]  
The SSIM is one of the most significant metrics for LDCT denoising as it considers human visual perception, focusing on structural similarity, luminance, and contrast. SSIM values range from 0 to 1, with values closer to 1 indicating high structural similarity to the reference image [22]. This is calculated as in Equation (4). The SSIM of denoised image (\(x\)) and the reference NDCT image (\(y\)) can be calculated using Equation (4), where \(\mu\) is the  
Figure 4: Edge enhancement at each upsampling and downsampling checkpoint labeled in Figure 2 using different filters in the attention module. (**a–c**) The feature maps generated from checkpoint at the encoder; (**d–f**) the feature maps generated from the checkpoint at the decoder. mean, \(\sigma\) is the sample deviation, \(\sigma_{xy}\) stands for the sample covariance, while \(\alpha,\beta,\gamma\geq 0\) and \(\alpha,\beta,\gamma>0\) are used for controlling the luminance, contrast, and structure similarities between the two images. The SSIM is particularly relevant in medical imaging because preserving anatomical structures is crucial for accurate diagnosis. Unlike the PSNR and MSE, the SSIM gives more weight to perceptual quality, making it a better metric for assessing how well a denoised image retains essential details. \[ \begin{split} SSIM(x,y)&=\frac{(2\mu_{x}\mu_{y}+c_{ 1})(2\sigma_{xy}+c_{2})}{(\mu_{x}^{2}+\mu_{y}^{2}+c1)(\sigma_{x}^{2}+\sigma_{y}^ {2}+c_{2})}\\ &=\left(\frac{2\mu_{x}\mu_{y}}{\mu_{x}^{2}+\mu_{y}^{2}}\right)^{ \alpha}\left(\frac{2\sigma_{xy}}{\sigma_{x}^{2}+\sigma_{y}^{2}}\right)^{\beta} \left(\frac{\sigma_{xy}}{\sigma_{x}\sigma_{y}}\right)^{\gamma}\end{split}  \tag{4}\]  
The standard error of the mean (SEM) can also be used to calculate the uncertainty of a result. This can be calculated using the expression in (5), where \(\sigma\) is the standard deviation, and \(\hat{N}\) is the mean [23]. \[SEM=\frac{\sigma}{\sqrt{\hat{N}}} \tag{5}\]  
The MSE calculates the average squared differences between predicted pixel values and target values, penalizing larger errors more heavily. This makes it highly effective for ensuring pixel-wise accuracy in reconstructed images. In the context of LDCT denoising, the MSE is straightforward to implement, calculated by applying Equation (6), and works well for minimizing noise while encouraging the model to match the intensities of the normal-dose reference image \(y_{i}\). However, the MSE focuses only on pixel differences and does not account for human perceptual quality, so it may lead to overly smooth results that miss important structural details.