## 2 Comparison Methods  
### Polynomial Fitting Method  
Right now, the feature-based methods (also known as polynomial fitting mapping methods) [40] use specific parameters to map the image feature points to screen gaze point coordinates (gaze point coordinates can be 2D or 3D coordinates). It avoids complete hardware calibration steps, and has better simplification and operation. Therefore, it is a commonly used method to solve the mapping relationship between data. Depending on the complexity of the model, there will be existed different forms of polynomials, such as polynomial of first degree, quadratic polynomial, and even other high-order polynomials. Here, it is assumed that the coordinates of the pupil center position in the eyeball coordinate system is \((x,y)\), and the gaze point coordinates in the scene plane coordinate system is \((u,v)\). This section mainly uses the classical quadratic polynomial mapping model as an example to explain its principle. In 1974, with continuous research [PERSON] et al. [41] finally proposed a video-based real-time gaze estimation method. The implementation process of this method mainly uses a pupil center and an optical lens combined with a photoelectric galvanometer to detect the gaze of human eye. They have proved the conclusion that the spatial distribution of pupil and gaze point is nonlinear through experimental research. Based on the above research, [PERSON] et al. [42] finally verified through a variety of polynomial experiments that the classic quadratic polynomial fitting method is more suitable for solving viewing direction of 2d. The expression is as follows:  
\[ \begin{split} u&=a_{0}x^{2}+a_{1}xy+a_{2}y^{2}+a_{3}x +a_{4}y+a_{5}\\ v&=b_{0}x^{2}+b_{1}xy+b_{2}y^{2}+b_{3}x+b_{4}y+b_{5} \end{split}  \tag{1}\]  
In the above formula, the eye feature vector is \(F_{e}=\left[x,y\right]^{T}\epsilon R^{2\times 1}\), Position vector of gaze point \(G_{g}=\left[u,v\right]^{T}\epsilon R^{2\times 1}\). [PERSON] uses one of the pupil corneal reflex methods, which requires a light source that forms a Purkinje image [43] through the corneal reflex, and the position of the Purkinje image on the cornea can be assumed to be stationary when head movements remain constant. With the Purkinje image as the datum, the drop of the sight on the screen can be obtained by bringing it into the mapping model. The pupil-corneal reflex method uses an infrared light source, and the phenomenon of \"bright pupil\" generated by it will make the video image easier to process. The production principle of bright pupil and dark pupil is shown in Figure 1.  
Theoretically, the higher the function order, the higher the fitting accuracy. Therefore, our previous work [31] is to study the polynomial fitting equation in depth, in order to make the data fitting uncomplicated, the higher order power is discarded and the fifth order term is retained, and propose a higher-order odd polynomial fitting equation, the expression is shown in (2). However, the actual accuracy is not very ideal. In this paper, we continue to study and a mapping equation based on homography is proposed. \[ \begin{split} u&=a_{0}x+a_{1}xy^{2}+a_{2}x^{3}+a_{3} xy^{4}+a_{4}x^{3}y^{2}+a_{5}x^{5}\\ v&=b_{0}y+b_{1}x^{2}y+b_{2}y^{3}+b_{3}x^{4}y+b_{4}x^{ 2}y^{3}+b_{5}y^{5}\end{split}  \tag{2}\]  
After the polynomial fitting method determining the mapping function equation, it needs to calibrate to solve the unknown parameters, \(a_{i}\) and \(b_{i}\) to obtain the appropriate gaze estimation regression model. The eye feature vector \(F_{e}\) is obtained from the pupil center coordinates and Purkinje image coordinates by the tester look directly at the screen and look at the calibration points at different positions in turn. The commonly used calibration points are 4, 5, 9 and 14. [PERSON] et al. [29] found that when the number of calibration points is 14, the calibration accuracy is the highest, followed by 9 points.