## 4 Experimental Design
### Evaluation Metrics  
To evaluate the performance of our LinearTGAN model, four standard SR metrics were employed: Peak Signal-to-Noise Ratio (PSNR) [35], Structural Similarity Index Measure (SSIM) [35], Learned Perceptual Image Patch Similarity (LPIPS) [36], and Frechet Inception Distance (FID) [37]. PSNR is a pixel-wise evaluation metric that measures the ratio between the maximum possible pixel value and the Mean Squared Error (MSE) [38] between the reference image and the generated image. It measures how closely the pixel intensities in the generated image match those of the reference image, with higher PSNR values indicating better image quality. SSIM evaluates image quality by comparing structural information, luminance, and contrast between the reference image and the generated image. While PSNR focuses on pixel-level accuracy, SSIM aligns more closely with human visual perception,as it considers the structural similarity of images. LPIPS measures perceptual similarity between the reference and generated images using deep feature representations extracted from pre-trained neural networks. This study utilized AlexNet [39] and VGG [40] as feature extraction networks. In contrast to PSNR and SSIM, which primarily address pixel-level and structural fidelity, LPIPS captures high-level perceptual differences, emphasizing semantic features and overall image realism. Lower LPIPS scores indicate a closer perceptual match between the generated and reference images. Lastly, FID quantifies perceptual quality by measuring the distributional similarity between feature representations of real and generated images. A lower FID score signifies reduced divergence between these distributions, indicating higher realism and improved generative performance. All metrics are defined mathematically as follows:  
\[\text{PSNR}(x,y)=10\cdot\log_{10}\left(\frac{\text{MAX}^{2}}{\text{MSE}(x,y)}\right) \tag{2}\]  
where MAX represents the maximum pixel intensity value and MSE denotes the mean squared error between two images (\(x\) and \(y\)). \[\text{SSIM}(x,y)=[L(x,y)]\cdot[C(x,y)]\cdot[S(x,y)] \tag{3}\]  
with  
\[L(x,y)=\frac{2\mu_{x}\mu_{y}+C_{1}}{\mu_{x}^{2}+\mu_{y}^{2}+C_{1}} \tag{4}\]  
\[C(x,y)=\frac{2\sigma_{xy}+C_{2}}{\sigma_{x}^{2}+\sigma_{y}^{2}+C_{2}} \tag{5}\]  
\[S(x,y)=\frac{\sigma_{xy}+C_{3}}{\sigma_{x}\sigma_{y}+C_{3}} \tag{6}\]  
where the luminance term (\(L(x,y)\)) quantifies the similarity in the mean intensity between images \(x\) and \(y\), the contrast term (\(C(x,y)\)) evaluates the similarity in contrast between the two images, and the structure term \(S(x,y)\), assesses their structural correspondence. Here, \(\mu_{x}\) and \(\mu_{y}\), \(\sigma_{x}^{2}\) and \(\sigma_{y}^{2}\), and \(\sigma_{xy}\) are the mean values, the variances, and the covariance between images \(x\) and \(y\), respectively. \[\text{LPIPS}(x,y)=\sum_{j}\frac{1}{W_{j}H_{j}}\sum_{h_{j}w}\
orm{\phi^{j}(x)- \phi^{j}(y)}_{2}^{2} \tag{7}\]  
with:  
\[\phi^{j}(x)=w_{j}\odot o_{lm}^{j}(x) \tag{8}\]  
\[\phi^{j}(y)=w_{j}\odot o_{lm}^{j}(y) \tag{9}\]  
where \(W_{j}\) and \(H_{j}\) denote the width and height of the feature map at layer \(j\), while \(o^{j}(x)\) and \(o^{j}(y)\) correspond to the internal activations of a pretrained CNN and the scaled channel-wise weight vector (\(w_{j}\)). \[\text{FID}(x,y)=||\mu_{x}-\mu_{y}||^{2}+\text{Tr}(\Sigma_{x}+\Sigma_{y}-2( \Sigma_{x}\Sigma_{y})^{\frac{1}{2}}) \tag{10}\]  
where \(\mu_{x}\) and \(\mu_{y}\) represent the mean feature vectors of image sets \(x\) and \(y\), respectively, while \(\Sigma_{x}\) and \(\Sigma_{y}\) denote the covariance matrices of feature activations for these image sets. The term \(||\mu_{x}-\mu_{y}||^{2}\) corresponds to the squared Euclidean distance [41] between the mean feature vectors, quantifying the shift in feature distributions.