## 6 Results
\begin{cases}l(\mu_{a}^{true},\mu_{a}^{recon})&=\frac{(\frac{2\mu_{ a}^{true}\mu_{a}^{recon}+C_{1}}{(\mu_{a}^{2}\mu_{a}^{recon}+\mu_{a}^{recon}+C_{1})} }{(\mu_{a}^{2}\mu_{a}^{recon}+\mu_{a}^{recon}+C_{1})}\\ c(\mu_{a}^{true},\mu_{a}^{recon})&=\frac{(\frac{\mu_{a}^{recon}\mu_{a} ^{recon}+C_{1}}{(\mu_{a}^{2}\mu_{a}^{recon}+\mu_{a}^{recon}+C_{2})}}{(\mu_{a} ^{2}\mu_{a}^{recon}+\mu_{a}^{recon}+C_{3})}\\ \end{cases} \]  
where \(l(\mu_{a}^{true},\mu_{a}^{recon})\), \(c(\mu_{a}^{true},\mu_{a}^{recon})\), and \(s(\mu_{a}^{true},\mu_{a}^{recon})\) are the luminance, contrast, and structure variations between the true image \(\mu_{a}^{true}\) and reconstructed image \(\mu_{a}^{recon}\), respectively, and \(x>0\), \(y>0\), and \(z>0\) are three parameters used to adjust relative importance of the three components of the similarity measure. \(\hat{\mu}_{\mu_{a}^{true}}\) and \(\hat{\mu}_{\mu_{a}^{recon}}\) are the means of pixel values of \(\mu_{a}^{true}\) and \(\mu_{a}^{recon}\), respectively. We denote by \(\sigma_{\mu_{a}^{true}}\), \(\sigma_{\mu_{a}^{recon}}\), and \(\sigma_{\mu_{a}^{true}}\), \(\mu_{a}^{recon}\) the standard deviation of \(\mu_{a}^{true}\) and \(\mu_{a}^{recon}\), and the covariance of image \(\mu_{a}^{true}\) and \(\mu_{a}^{recon}\), respectively. \(C_{1}\), \(C_{2}\), and \(C_{3}\) are constants. The global comparison of quality of reconstructed images are shown in Figure 7a,b, as the result of running the Kruskal-Wallis for PSNR (we eliminated AmsGrad outliers where \(PSNR<-25\) db) and SSIM, grouping each score sample by optimizer. From Figure 7, it appears that the PSNR and the  
Figure 6: Correlation between number of iteration of each optimizer and (**a**) initial guess error \(\delta\) (**b**) learning rate hyper parameter \(\beta\). SSIM of AmsGrad are much lower (worse quality) than those of Nadam and Adam. In addition, we can observe that the means of PSNR and SSIM for Adam and Nadam are very close. To evaluate the influence of number of inclusions on image quality, we conduct a Wilcoxon test [25]. The test was applied according to different groups of numbers of inclusions. The resulting \(p\)-values of this test are summarized in Table 6. The results analysis shows that there is a significant statistical difference between means due to the difference in number of inclusion present in images (\(p\)-value \(<0.05\)). A similar conclusion is deduced about the influence of learning rate on PSNR and SSIM. Scatter plots in Figure 8a,b clearly show this strong influence of learning rate on PSNR and SSIM, respectively. The resulting [PERSON]'s correlation coefficients by optimizer (and the corresponding \(p\)-value) for PSNR and SSIM are mentioned at the top of each graph. As shown in Figure 8a, we notice that there is a strong negative correlation between learning rate hyper-parameter and PSNR for the case of Adam and AmsGrad. For the case of Nadam, we note a moderate negative correlation between the choice of learning rate and PSNR of reconstructed images. From Figure 8b, we observe that there is a strong negative correlation between learning rate parameter and SSIM for the case of [PERSON] and Nadam. In addition, there is a moderate negative correlation between learning rate and SSIM in the case of AmsGrad. The resulting \(p\)-values mentioned at the top of each graph indicate that these correlations are statistically significant, and, consequently, we can conclude the same about the significance of the influence of learning rate choice on the quality of the resulting reconstructed image. Thus, a small value of learning rate that ranges between 0.001 and 0.2 is recommended. Concerning initial guess error, scatter plots in Figure 9 demonstrate the influence of initial guess error on reconstructed image quality. From Figure 9a,b, the obtained results show that there is no significant statistical differences between the initial guess error and resulting quality (PSNR/SSIM).