## 2 Comparison Methods
### Homography Normalization  
The hardware system based on the method of homography normalization [39] is configured as a camera and four light sources placed at the four corners of the screen. In this method, gaze point is estimated by the transformation relationship between two projections. The basic principle is to calculate the corresponding two homography matrices according to the projection mapping relationship between the imaging plane and the corneal reflection plane, and between the corneal reflection plane and the screen plane. When estimating the sight drop point, according to the position of the pupil center on the imaging plane, the homography matrix \(H_{1}^{N}\) formed by the imaging plane and the corneal reflection plane is used to convert the point \(P\) on the corneal reflection plane. However, since the system is not punctuated, the 3D coordinates of the corneal reflex plane are unknown. Therefore, it is assumed that the corneal reflex plane is a normalized plane, i.e., a unit square with  
Figure 1: Pupil formation principle of light pupil and dark pupil [44].  
predefined coordinates. Then, the homography matrix \(H_{S}^{N}\) formed by the corneal reflection plane and the screen plane is converted into the point S on the screen plane, which is the fixation point. \(H_{1}^{N}\) can be calculated from four CRs points generated by corneal reflex and the four corner points of predefined normalized plane. The projection matrix \(H_{S}^{N}\) composed of normalized plane and screen plane can be determined through the calibration process, and finally the projection transformation matrix from imaging plane to screen can be obtained:  
\[H_{1}^{S}=H_{1}^{N}H_{N}^{S} \tag{3}\]  
Since the hardware condition based on the homography normalization method must be one camera and four light sources, there are many improvement methods for this problem. [PERSON] (2016) proposed the method by proposing three optional geometric transformations of corneal reflection plane, all of which are adaptive, enabling the examination to be carried out with two or three CRs and improved accuracy. [PERSON] (2016) can predict the change of homography when the head is in a new position by simulating head changes, so as to calibrate, thus improving the robustness of eye tracking.  
The method proposed in this paper is different from the Homography Normalization method. In the coordinate system of 3D world, we use eye coordinate system, imaging plane coordinate system and screen coordinate system to establish the mapping relationship between imaging plane coordinates and pupil center coordinates, and between pupil center coordinates and screen gaze point coordinates through geometric relationship, thus forming the mapping equation based on homography transformation. In the relationship between the imaging plane coordinates and the pupil center coordinates, the pupil center coordinates are obtained by using the camera model(pin-hole imaging). Then, in the relationship between the pupil center coordinates and the screen gaze point coordinates, we use the similarity conditions of similar triangles to calculate the gaze point coordinates. In the homography normalization method, a normalized plane is constructed on the pupil through the four points of the Purkinje image to obtain two mapping matrices, so as to calculate the gaze point position, four CRs points are the basic condition. Although the improved method based on homography normalization method can improve the robustness, it also increases the complexity of the algorithm. Our proposed method can obtain the results only through simple mathematical geometric relationship. From the perspective of hardware, the proposed method only needs a camera and an infrared light to obtain the bright pupil image, instead of using four infrared lights to obtain four CRs points. To sum up, the proposed method has higher accuracy and reduces the cost of hardware, and the algorithm is more simple and feasible.