## 2 Methodology
### Performance Evaluation
#### 2.4.1 Image Quality Evaluation  
Many metrics are available to evaluate the quality of artificial images, and each has strengths. However, interpreting the results can be challenging and requires careful consideration. For example, a specific FID variation (Gromov-Frechet Distance) was  
Figure 4: Example images from the LG dataset: (**a**,**b**) male and (**c**,**d**) female. Figure 5: Example images from the UCSB dataset: (**a**,**b**) benign tumors, (**c**,**d**) malignant tumors. Figure 3: Example images from the LA dataset: (**a**,**b**) benign tumors, (**c**,**d**) malignant tumors. Figure 2: Example images from the CR dataset: (**a**,**b**) benign tumors, (**c**,**d**) malignant tumors. designed to compare metric space shapes, particularly in shape analysis and topology. It has valuable applications such as shape matching or comparing complex geometric objects. However, GAN-generated images are typically evaluated based on the similarity of their distributions to authentic images, where the focus is on pixel values, textures, and high-level features captured by neural networks rather than the geometric structure of the underlying data. Some assessments could also be performed via precision-recall (PR) metrics. These metrics are derived from the F1 score or area under the PR curve and can provide insights into a GAN's performance in generating realistic and diverse images. For instance, PR metrics can provide a valuable perspective by assessing the trade-off between the precision (how many of the generated images are relevant or high-quality) and recall (how many of the relevant images are generated) of the model. However, some disadvantages compared to FID and IS are observed. PR metrics require the definition of thresholds to determine what constitutes a realistic or diverse image, which can be complex and context-dependent. The choice of threshold can significantly affect the results, making PR metrics less consistent across different models and datasets. The interpretation of PR curves might be more complex than single-valued metrics like FID or IS. The FID and IS are particularly pertinent for assessing image quality because they have been widely adopted in the field and are well-suited for comparing generative models by evaluating the fidelity and diversity of the generated images. FID, for instance, compares the distribution of generated images with real images, capturing the similarity in a way that aligns with human perception. IS measures how distinct and meaningful the generated images are using the classification confidence of a pre-trained network. In this work, we applied the FID metric [38] to assess the quality of artificial images quantitatively. This metric measures the distance between the distributions of real and generated images. Thus, lower FID scores indicate higher similarity between the distributions, meaning that the generated images closely resemble the original ones. The FID measures the similarity between two multivariate Gaussian distributions, defined by the mean and covariance matrix of activation features extracted from Inception v3's 2048 th layer. Mathematically, the FID score is defined by  
\[FID=\|\mu_{\tau}-\mu_{t}\|^{2}+\text{Tr}(\Sigma_{\tau}+\Sigma_{\text{f}}-2( \Sigma_{\tau}\cdot\Sigma_{\text{f}})^{0.5}), \tag{16}\]  
where \(\mu_{\tau}\) and \(\mu_{t}\) are the mean features of real and fake images. \(\Sigma_{\tau}\) and \(\Sigma_{\text{f}}\) are the covariance matrices of real and fake image features, and \(\text{Tr}(\cdot)\) denotes the trace of a matrix. We also applied the IS metric [39] to estimate the diversity of the generated images. A higher IS suggests greater variety in the assigned classes, although it does not necessarily indicate a high degree of realism. In the IS calculation, fake images were evaluated based on the activations of the final classification layer of a pre-trained Inception v3 model. This model assigns a probability distribution to each image over predefined classes in the ImageNet dataset.