# Neural Network-Assisted Interferogram Analysis Using Cylindrical and Flat Reference Beams
## 4 Discussion  
In this section, we discuss the relationship between an increase in the sensitivity of interferograms due to the structural complexity of the reference beam and a decrease in MAE recognition by the neural network. In order to quantitatively estimate the sensitivity of recognition of the interferogram under consideration, we calculate the values of the parameters \(S_{1}\) and \(S_{2}\), which correspond to the value (9) of the RMSE of the \(p\)-type interferogram formed by an aberrated wavefront \(I_{p}(x,y)\), from the interferogram corresponding to the absence of aberrations \(I_{0p}(x,y)\). Figure 12 shows the RMSE values (9) of standard (without aberrations) cylindrical interferograms compared to aberrated ones. On average, the RMSE has a uniform dis \begin{table}
\begin{tabular}{c c c c} \hline \hline  & \multicolumn{3}{c}{**Images of Cylindrical Interferograms**} \\  & \multicolumn{3}{c}{\(D_{20}Z_{20}(x,y)+D_{44}Z_{44}(x,y)\)} \\ \hline \(D_{20}\) & & 0.30 \\ \(D_{44}\) & 0.05 & 0.15 \\ \(D_{44}\) & & 0.25 \\ _Interf._ & & \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Examples of experimentally obtained linear and cylindrical interferograms for various types and levels of aberrations. Figure 11: The neural network learning process dynamics using experimental cylindrical interferograms with the parameters \(\alpha=2\pi\) (a), \(\alpha=4\pi\) (b), and \(\alpha=6\pi\) (c) expressed by the dependence of MAE on the number of epochs. \begin{table}
\begin{tabular}{c c c} \hline \hline \multirow{2}{*}{**Type Angle**} & **Cylindrical Model** & **Cylindrical Experimental** \\  & **Interferogram** & **Interferogram** \\ \hline \(\alpha=2\pi\) & 0.0050 & 0.0049 \\ \(\alpha=4\pi\) & 0.0047 & 0.0046 \\ \(\alpha=6\pi\) & 0.0050 & 0.0052 \\ \hline \hline \end{tabular}
\end{table} 
Table 8: Mean absolute recognition error for cylindrical interferograms from the model and experimental dataset. tribution among the first 8 types of aberrations, except for radially symmetric ones. For aberrations of the \(Z_{20}\) ii \(Z_{40}\) types, corresponding to defocusing and spherical aberration, an almost twofold increase in the RMSE is observed over the entire range of \(c_{mn}\) from 0.1 to 1. Figure 13 shows the average values of the RMSE of linear and cylindrical interferograms. It has been found that the RMSE increases with an increase in the aberration level \(c_{mn}\) and barely depends on the tilt angle of the reference beam \(\alpha\). However, for cylindrical interferograms, the best sensitivity is at \(\alpha=4\pi\)-\(6\pi\), while for a flat reference beam at \(\alpha=2\pi\). This narrows the applicability range of a flat reference beam. In addition, at levels of aberration \(c_{mn}\geq 0.5\), the sensitivity of cylindrical interferograms is higher than that of linear ones. For clarity, the value \(\Delta\) is introduced--the sensitivity of interferogram recognition by machine learning methods (11): if \(\Delta\geq 0\), then the sensitivity of the cylindrical interferogram is higher than the linear one, otherwise the sensitivity is lower. Figure 14 shows diagrams of the average sensitivity for various types of aberrations (\(Q=8\)) in cylindrical and linear interferograms with variations in the parameters \(\alpha\) and \(c_{mn}\). It can be seen from the graphs in Figure 14 that, for small aberrations, the cylindrical interferogram is less sensitive than the linear one. The cylindrical interferogram is more sensitive than the linear one for aberrations with levels of \(c_{mn}>0.3\). Figure 12: RMSE values of cylindrical interferograms (\(\alpha=2\pi\)) for different types of aberrations \(Z_{mn}\). Figure 13: Average RMSE values for the first 8 types of aberrations for linear (**a**) and cylindrical (**b**) interferograms (\(\alpha=2\pi\dots\ 10\pi\)).