## 3 Method
### Build the Mapping Equation
In addition, it is considered that the verification is performed outside \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline
**Fixation Point** & **1** & **2** & **3** & **4** & **5** & **6** & **7** & **8** & **9** & **Accuracy** \\ \hline experiment 1 (+(blue))\({}^{\circ}\) & 0.733 & 0.114 & 0.182 & 0.348 & 0.388 & 0.872 & 0.898 & 0.585 & 0.469 & - \\ \hline experiment 2 (+(red))\({}^{\circ}\) & 0.387 & 0.292 & 0.371 & 0.087 & 0.354 & 0.554 & 0.257 & 0.379 & 0.287 & - \\ \hline mean value & 0.560 & 0.202 & 0.276 & 0.218 & 0.371 & 0.713 & 0.577 & 0.482 & 0.378 & 0.420 \\ \hline variance & 0.060 & 0.016 & 0.018 & 0.034 & 0.001 & 0.050 & 0.206 & 0.021 & 0.017 & 0.047 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Error analysis of two experimental results in the nine-point calibration method (in degrees). Figure 7: Nine-point verification results of the mapping equation. Secondly, F is F-statistic for short. Generally speaking, it tests the significance of the entire equation. The formula is expressed as the sum of squares divided by the sum of squared errors. This can indicate that the larger the mean square error of the classic quadratic polynomial mapping equation and the smaller the F value, and the lower the confidence of the estimated value. Finally, we perform residual map regression plots on the two mapping models and judge whether there are exist abnormal points in the calibration data set. Among them, we only do regression diagnosis for abscissa of the two methods, and the analysis results of ordinate are similar. The results are shown in Figure 12.  
The main concepts involved in the 2d residual graph include: shape, amplitude, residual value, and confidence interval. The residual graph mainly depends on whether the shape of the experimental results is within the specified range and the magnitude of its amplitude is not specified. It can be analyzed from the residual diagram that when the confidence intervals of the residuals (the vertical lines pass through the zero point) contains zero points, it indicates that the regression model fits well, otherwise the worse. The nine green circles in the figure represent the residual value, and the nine vertical lines represent the range of the residual confidence interval. If the confidence interval basically passed through the origin, it means that the equation fits well, otherwise, the fitting is poor. As can be seen from Figure 12, the ninth gaze point in quadratic polynomial fails to pass through the origin, and it is a red vertical line, indicating that this point is an abnormal point, which will lead to poor fitting effect. After many times of non-linear fitting of the data and analysis of the residual plot, the abscissa and ordinate residual plots in the classic quadratic polynomial model will appear abnormal points. The mapping equation based on the homography transformation fitting surface and residuals plot are basically kept in a stable range, and the model fitting effect well. Therefore, the prediction accuracy of this method is higher than the classical quadratic polynomial mapping equation. By consulting the literature, we compare the accuracy of our method with other methods. The comparison results are shown in the table below. From Table 3, we can see that our method has simple hardware and high accuracy, so it shows that our method can achieve good experimental accuracy. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Method** & **Number of Camera** & **Number of Light Sources** & **Accuracy** \\ \hline Our method & 1 & 1 & 0.5\({}^{\circ}\) \\ \hline [PERSON] [42] & 1 & 1 & 1.23\({}^{\circ}\) \\ \hline [PERSON] [46] & 1 & 4 & 0.72\({}^{\circ}\)\(\pm\) 0.35\({}^{\circ}\) \\ \hline [PERSON] [39] & 1 & 4 & 1.03\({}^{\circ}\)–1.17\({}^{\circ}\) \\ \hline [PERSON] [36] & 1 & 5 & 0.7\({}^{\circ}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Comparison of experimental accuracy of different methods. Figure 12: Schematic diagram of gaze point ordinate surface fitting.