## 2 Materials and Methods
### Evaluation Metrics  
#### 2.4.1 Evaluation Metrics for Image Generation Tasks  
For image generation tasks, we use Inception Score (IS), Kernel Inception Distance (KID), and Learned Perceptual Image Patch Similarity (LPIPS) as metrics [46]. IS evaluates the diversity and authenticity of the generated images by calculating the KL divergence between the distributions of the model-generated images and the real images. A high IS value indicates the diversity and high quality of the generated images. The calculation formula is as follows:  
\[\text{IS}=\exp\bigl{(}E_{x\sim p_{t}}[D_{KL}(p(y|x)||p(y))]\bigr{)} \tag{8}\]  
Here, \(p_{t}\) represents the distribution of the actual data, \(p(y|x)\) represents the conditional probability distribution of predicting category \(y\) given input \(x\), \(p(y)\) represents the marginal probability distribution of the model predicting category \(y\) over the entire dataset, and \(D_{KL}\) is the Kullback-Leibler divergence, which measures the difference between two probability distributions.  
Figure 4: Images generated by the StyleGAN2-ADA model using consecutive-layer 2D CT grayscale images as training data exhibiting mode collapse.  
Figure 3: Consecutive-layer 2D CT grayscale images of the first sandstone column sample.  
Due to the inherent bias affecting the evaluation results of the Frechet Inception Distance (FID) when there are fewer real image samples, FID is not suitable as an evaluation metric for our image generation task [47]. To address this issue, we have adopted the unbiased Kid as an alternative. Kid assesses the performance of the generative model by measuring the difference between the mean embeddings of the real data and generated data in the feature space. The formula is as follows:  
\[\text{KID}=\sqrt{\textit{Tr}\Big{(}(\Phi_{t}-\phi_{m})^{\textit{T}}(\phi_{t}- \phi_{m})\Big{)}} \tag{9}\]  
\(\phi_{t}\) represents the embedded representation of the real dataset, typically extracted through a specific layer of the Inception network, while \(\phi_{m}\) is the embedded representation of the image output by the generative model, also extracted through the same layer of the Inception network. \(\sqrt{\textit{Tr}(A^{\textit{T}}A)}\) is the Hilbert-Schmidt norm, which can be used to measure the difference between two embedded representations.  
LPIPS is an advanced image quality assessment tool that uses deep learning models to measure the perceptual differences between images. Compared to traditional pixel-level comparison methods such as SSIM, LPIPS is more reflective of human visual perception characteristics. In the LPIPS assessment process (see Figure 5), the input of two images is first passed through a pre-trained convolutional neural network (such as VGG19) to extract feature embeddings, which capture high-level semantic information of the images. Subsequently, the feature embeddings are normalized on each channel, scaled to the range of [\(-\)1, 1] or [0, 1], and then weighted by a weight vector. Next, the L2 distance of the weighted feature embeddings is calculated to quantify the differences in the feature space of the images. The L2 distances of all the layers are averaged to obtain a global distance measurement, corresponding to the d0 and d1 of the image pair, respectively. Finally, these two sets of global distance measurements are used as inputs to provide a small network G, which is trained to predict human perceptual judgments of the image pairs, outputting h.