## 4 Results and Discussion  
In this section, we present the results of an experiment on the proposed system using 5000 pairs of input and teaching data generated by numerical simulations. We generated teaching data in the form of \(64\times 64\)-pixel images deformed by changing the attributes of each oval, i.e., its intensity, size, position, and rotation angle, from the Shepp-Logan phantom [54] shown in Figure 4a. The attributes were varied in the ratio ranges shown in Table 1 from that of the base phantom by using uniform random numbers. The MATLAB (MathWorks, Natick, MA, USA) function phantom [55] generated the images. To avoid generating extremely large or small ovals that should not be considered, some common deformation rates were chosen for one attribute in some of the sets of ovals in one phantom. The input data were projections calculated by Equation (1) by using a common projection operator \(A\), projection noise \(\delta\), and the target image \(x\) as the corresponding teaching data. The projection operator \(A\), represented as a matrix, was simulated on the basis of the Radon transform assuming parallel projections from 90 directions with 95 detectors, so \(I=90\times 95\) and \(J=64\times 64\). The signal-to-noise ratio (SNR) for each projection was selected so as  
Figure 1: Structure of the whole network \(\Theta\). Figure 3: Structure of block \(p\). Figure 2: Structure of the optimizeee \(\Omega\). to be uniformly distributed in the range of 20 dB to 50 dB. Figure 5 shows the resulting experimental histogram of SNRs, while Figure 6 shows samples of learning data generated by the above procedure. In the experiment, we designed the optimizer \(\Lambda\) to be a simple and typical network (Figure 7). The batch normalization layer [56] is supposed to stabilize learning, reduce the initial value dependence of the learning parameters, and speed up learning. The dropout layer [57] is used to suppress overfitting. Since the first hidden layer is dense, it requires a massive number of training parameters, resulting in a vast computational cost for training. Therefore, designing a more efficient optimizer \(\Lambda\) is a future challenge. The parameters \(\gamma\) and \(\alpha\) are power exponents, and giving them excessive values makes the numerical operations unstable. However, in the process of automatically updating the learning parameters by using the backpropagation procedure, \(\gamma\) and \(\alpha\) are given large values that temporarily exceed the tolerance of a stable calculation. We avoided this problem by setting an upper limit on the output values of the optimizer \(\Lambda\). \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multirow{2}{*}{**Group**} & **Oval** & \multirow{2}{*}{**Intensity**} & **Minor Axis** & **Major Axis** & **Horizontal** & **Vertical** & **Angle of** \\  & **Numbers** & & **Length** & **Length** & **Position** & **Position** & **Rotation** \\ \hline
1 & \#1, \#2 & \(\pm\)10\% & \(\pm\)20\% * & \(\pm\)15\% * & \(\pm\)0\% & \(\pm\)0\% & \(\pm\)0\% \\
2 & \#3, \#4, \#5 & \(\pm\)30\% & \(\pm\)30\% & \(\pm\)20\% & \(\pm\)25\% & \(\pm\)25\% & \(\pm\)20\% \\
3 & \#6, \#7 & \(\pm\)30\% & \(\pm\)50\% & \(\pm\)50\% & \(\pm\)50\% & \(\pm\)50\% & \(\pm\)50\% \\
4 & \#8, \#9, \#10 & \(\pm\)30\% * & \(\pm\)25\% * & \(\pm\)25\% * & \(\pm\)20\% * & \(\pm\)20\% * & \(\pm\)50\% \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Deformation ratios from base phantom. Symbol * indicates that the deformation ratio for the attribute uses a common value for each oval in the same group. See Figure 4b for “oval numbers”. Figure 4: Base phantom. Figure 5: Histogram of projection SNRs in input data set. We define the activation function of the output layer of \(\Lambda\) to be  
\[h(x_{1},x_{2}):=1.8\times(\text{sgm}(x_{1}),\text{sgm}(x_{2}))^{\top}, \tag{11}\]  
where the function sgm is the standard sigmoid function.