# Two-View Mammogram Synthesis from Single-View Data Using Generative Adversarial Networks
## 2 Materials and Methods
### Performance Evaluation  
We evaluated the performance of CR-GAN and our proposed models in terms of the image similarity of each pair between the real and synthesized images. The Frechet inception distance (FID) has been widely used for performance analyses of GAN models [42; 43]. [PERSON] et al. have used the sliced Wasserstein distance (SWD) to assess the performance of PG-GAN [23]. Nonetheless, these metrics measure distances between synthetic and real data distributions; that is, the FID and SWD evaluate not the similarity between each real and fake sample, but the entire similarity between the two groups. Thereby, we used the peak signal to noise ratio (PSNR), structural similarity (SSIM) [44], multi-scale SSIM (MS-SSIM) [45], and cosine similarity (Cos_sim) to evaluate the similarity between pairs of real and synthesized CC-view mammograms for the test data set described in Section 2.2.  
The PSNR measures the image similarity based on the ratio of noise to the maximum pixel value, as follows:  
\[PSNR=10\log_{10}\left(\frac{P_{max}}{MSE}\right), \tag{9}\]  
where MSE is the mean square error (MSE) between two images, and \(P_{max}\) is the maximum value of the image pixels. If the images have 8-bit contrast resolution, \(P_{max}\) is 255. The higher the PSNR value, the more similar the two images.  
SSIM computes the image similarity between two images (**x,y**) using the three components of brightness, contrast, and structure, as shown in Equation (10):  
\[SSIM(\textbf{x},\textbf{y})=[l(\textbf{x},\textbf{y})][c(\textbf{x},\textbf{y })][s(\textbf{x},\textbf{y})], \tag{10}\]  
where \(l(\textbf{x},\textbf{y})\) is the brightness comparison, \(c(\textbf{x},\textbf{y})\) is the contrast comparison, and \(s(\textbf{x},\textbf{y})\) is the structural comparison [44]. The respective components are calculated according to the means, variances, and covariances of **x** and **y**. The SSIM value ranges between 0 and 1; as the similarity increases, the value becomes closer to 1.  
MS-SSIM has been introduced as an alternative metric of SSIM, incorporating image details at various resolutions [45]. MS-SSIM is calculated by combining the three components of SSIM on multiple scales, as follows:  
\[SSIM(\textbf{x},\textbf{y})=[l_{M}(\textbf{x},\textbf{y})]^{\alpha_{M}}\cdot \prod_{j=1}^{M}[c_{j}(\textbf{x},\textbf{y})]^{\beta_{j}}[s_{j}(\textbf{x}, \textbf{y})]^{\gamma_{j}}. \tag{11}\]  
The two images (**x,y**) are iteratively low-pass filtered and down-sampled by a factor of 2. The scale of the original image is 1, while that of the most reduced image is \(M\). The brightness comparison is computed only at scale \(M\), and we refer to this as \(l_{M}(\textbf{x},\textbf{y})\). The contrast and structure comparison components are calculated at each scale, denoted as \(c_{j}(\textbf{x},\textbf{y})\) and \(s_{j}(\textbf{x},\textbf{y})\) for the \(j^{\text{th}}\) scale, respectively. [PERSON] et al. [45] have obtained five-scale parameters, in which the SSIM scores agreed with subjective assessments. The resulting parameters were \(\beta_{1}=\gamma_{1}=0.0448\), \(\beta_{2}=\gamma_{2}=0.2856\), \(\beta_{3}=\gamma_{3}=0.3001\), \(\beta_{4}=\gamma_{4}=0.2363\), and \(\alpha_{5}=\beta_{5}=\gamma_{5}=0.1333\). We also used these parameters in our performance evaluation.  
Cos_sim is commonly used to determine the similarity between two vectors [46]. In some self-supervised contrastive learning methods (which have developed rapidly in recent years), Cos_sim has been utilized in classification tasks by placing similar images closer and dissimilar images further from each other in the latent space [46; 47]. Cos_sim is defined in terms of the cosine of the angle between the two vectors, and is calculated as follows:  
\[Cos\_sim(\textbf{x},\textbf{y})=\frac{\textbf{x}\cdot\textbf{y}}{||\textbf{x }||||\textbf{y}||}. \tag{12}\]_Cos_sim_ takes values within the interval from \(-1\) to \(1\). When two images are more similar, the _Cos_sim_ value is closer to \(1\).