## 6 Results
Thus, we can conclude with high confidence (\(p>0.05\)) that the image quality is only influenced by the number of anomalies in the image and the choice of the learning rate. We illustrate some cases from our simulation. Figure 10 shows the reconstructed absorption coefficient \(\mu_{a}\) for the case of one inclusion for an initial guess error equal to \(\delta=0.2\). Different values  
\begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline  & \multicolumn{3}{c}{**Adam**} & \multicolumn{3}{c}{**Nadam**} & \multicolumn{3}{c}{**AmsGrad**} \\ \hline  & **1 vs. 2** & **2 vs. 3** & **1 vs. 3** & **1 vs. 2** & **2 vs. 3** & **1 vs. 3** & **1 vs. 2** & **2 vs. 3** & **1 vs. 3** \\ \hline PSNR & 0.02 & \(2.5\times 10^{-4}\) & \(1.8\times 10^{-6}\) & 0.002 & 0.004 & \(7.8\times 10^{-11}\) & 0.003 & 0.031 & 0.001 \\ \of learning rate are used. The background of true images are taken equal to \(\mu_{a}^{bck}=0.01\) mm\({}^{-1}\) and \(\mu_{s}^{bck}=2\) mm\({}^{-1}\). The reconstruction using Nadam and Adam showed a good localization of inclusion. In addition, its size is the same compared to the true image with optical properties close to those of true image values. Some artifacts are observed in the borders close to sources and detectors region when the learning rate is higher than 0.1. For the case of AmsGrad reconstruction, we observe that the size of reconstructed image matches those for the true image with some artifacts in the center when the learning rate is lower than 0.1. However, when the learning rate is greater than 0.1, we remark that AmsGrad can localize the inclusion, but with some artifacts in the borders. The size and the shape of inclusion do not match those in the true image. Figure 11 shows the reconstructed absorption coefficient \(\mu_{a}\) for the case of two inclusions with different shapes for the same values of initial guess error and optical properties used in the first case of one inclusion. From Figure 11, we notice that we obtain a good localization of both inclusions for the case of Nadam and Adam for different values of learning rates. However, when the learning rate is higher than 0.01, we observe some artifacts near the borders. For the case of AmsGrad reconstruction, it is clear that, when the learning rate exceeds 0.1, the size and the shape of inclusion do not match those figuring in the true image. Figure 10: Reconstruction of the absorption coefficient \(\mu_{a}\) with one inclusion. The first row presents the true image (**left**) and initial guess image with an initial guess error \(\delta=0.2\) (**right**). The second, third, fourth, and fifth rows present the reconstruction images using the learning rate \(\beta\) taking values equal to 0.001,0.01,0.1, 0.2, and 0.3, respectively, using Nadam, Adam, and AmsGrad, from (**left** to **right**). Figure 11: Reconstruction of the absorption coefficient \(\mu_{a}\)with two inclusions. The first row presents the true image (**left**) and initial guess image with an initial guess error \(\delta=0.2\) (**right**). The second, third, fourth, and fifth rows present the reconstruction images using the learning rate \(\beta\) taking values equal to 0.001,0.01, 0.1, 0.2, and 0.3, respectively, using Nadam, Adam, and AmsGrad, from (**left to right**).