## 3 Comparison of Image Estimation Functions
\begin{table}
\begin{tabular}{c c c c c c c c} \hline
**Scene a** & **Scene b** & **Scene c** & **Scene d** & **Scene e** & **Scene f** & **Scene g** & **Scene h** \\ \hline
0.6642 & 0.9658 & 0.8443 & 0.7425 & 0.8892 & 0.8442 & 0.7559 & 0.9536 \\
0.7354 & 0.9904 & 0.8651 & 0.7935 & 0.9117 & 0.8639 & 0.8248 & 0.9810 \\
0.8543 & 1.0000 & 0.9318 & 0.9051 & 0.9610 & 0.9317 & 0.9146 & 0.9983 \\
1.0000 & 0.9835 & 1.0000 & 1.0000 & 1.0000 & 1.0000 & 1.0000 & 1.0000 \\
0.8523 & 0.9734 & 0.9073 & 0.8754 & 0.9626 & 0.9252 & 0.9236 & 0.9881 \\
0.7332 & 0.9256 & 0.8426 & 0.7640 & 0.9173 & 0.8631 & 0.8648 & 0.9689 \\
0.6942 & 0.9052 & 0.7932 & 0.7223 & 0.8246 & 0.7980 & 0.7226 & 0.9677 \\ \hline \end{tabular}
\end{table} 
Table 4: The change of the Laplace function during focusing in different scenes. \begin{table}
\begin{tabular}{c c c c c c c c} \hline
**Scene a** & **Scene b** & **Scene c** & **Scene e** & **Scene f** & **Scene g** & **Scene h** \\ \hline
0.8253 & 0.9182 & 0.8879 & 0.8534 & 0.8842 & 0.8842 & 0.8994 & 0.9681 \\
0.8525 & 0.9342 & 0.9038 & 0.8842 & 0.9234 & 0.9055 & 0.9146 & 0.9842 \\
0.9715 & 0.9832 & 0.9437 & 0.9714 & 0.9389 & 0.9593 & 0.9762 & 0.9973 \\
1.0000 & 1.0000 & 1.0000 & 1.0000 & 1.0000 & 1.0000 & 1.0000 & 1.0000 \\
0.9194 & 0.9741 & 0.9194 & 0.9342 & 0.9734 & 0.9294 & 0.9537 & 0.9748 \\
0.8621 & 0.9325 & 0.8856 & 0.8792 & 0.863 & 0.8913 & 0.8774 & 0.9695 \\
0.8423 & 0.8832 & 0.8723 & 0.8344 & 0.7841 & 0.8725 & 0.8122 & 0.9599 \\ \hline \end{tabular}
\end{table} 
Table 3: The change of the variance function during focusing in different scenes. Table 5 lists the entropy function evaluation values of the focus position and the position of the front and back three steps in eight scenes. The fourth row of data presents the focus position function value. For all these scenes, the sharpness estimations by the aforementioned IEFs of the captured image curves were calculated and are shown in Figure 4. In order to analyze the performance of each IEF, a whole-area focusing window was adopted, i.e., the whole picture was involved in the image estimation. For each subfigure in Figure 4, the abscissa and the ordinate are the picture sequence (where the ninth is the correct focus of the picture) and the normalized estimation function value, respectively. In Figure 4a-h, the entropy function has multiple peaks, indicating the presence of instability. In Figure 4h, the Brenner function and the Laplace function also have multiple extrema which are not applicable. In terms of the unimodal property found in the corresponding eight scenarios, the Tenengrad and the variance IEFs are relatively stable. In addition, the normalized Tenengrad IEF value drops faster on both sides of the peak, indicating that the Tenengrad IEF has higher sensitivity than the variance one. In order to quantify the focusing effect of different IEFs, the efficiency factors was defined as  
\[\text{Ef}=\frac{E_{max}-E_{min}}{E_{max}+E_{min}} \tag{10}\]  
where \(E_{\text{max}}\) and \(E_{\text{min}}\) represent the maximum and minimum values of the IEF, respectively. The higher the value of Ef is, the better the focusing effect is. Table 6 lists the Ef values of the different IEFs in the eight scenarios. \begin{table}
\begin{tabular}{c c c c c c} \hline  & **Tenengrad** & **Brenner** & **Variance** & **Laplace** & **Entropy** \\ \hline Scene a & 0.371 & 0.156 & 0.131 & 0.250 & 0.027 \\ Scene b & 0.212 & 0.204 & 0.183 & 0.136 & 0.050 \\ Scene c & 0.273 & 0.104 & 0.111 & 0.156 & 0.049 \\ Scene d & 0.326 & 0.211 & 0.161 & 0.244 & 0.035 \\ Scene e & 0.289 & 0.215 & 0.214 & 0.240 & 0.080 \\ Scene f & 0.333 & 0.112 & 0.139 & 0.241 & 0.085 \\ Scene g & 0.321 & 0.186 & 0.183 & 0.293 & 0.063 \\ Scene h & 0.083 & 0.092 & 0.058 & 0.061 & 0.032 \\ \hline \end{tabular}
\end{table} 
Table 6: Ef values of different IEFs in the eight scenarios.