# Presenting a Human Pupil Orbit Model (HPOM) for Eye-Gaze Tracking
## 2 Materials and Methods
### HPOM Fitting  
The accuracy of the HPOM estimated according to the proposed theory is proportional to the error of the RAE obtained by analysing the PPC. To improve accuracy and stability, HPOM fitting is performed using the method proposed in this paper on the RANdom SAmple Consensus (RANSAC) published by [PERSON], [PERSON] and [PERSON] [33]. The RANSAC method has a feature of repeatedly performing mathematical model estimation by randomly selecting the model Estimation Sample (ES) from the Total Sample (TS) and deriving an optimal result. As a result, even if the TS contains error samples, it shows high accuracy. However, there is a disadvantage in that the computational load is large in proportion to the number of TSs and ESs. Therefore, it is necessary to set the appropriate number of TS and ES to minimize the computational load with a good performance to derive an optimal model.  
HPOM fitting repeats the hypothesis and verification steps according to the sequence shown in Figure 9. In the hypothesis stage, ES is randomly selected from TS and HPOM is estimated according to the proposed theory. The number of straight lines that can be calculated in the ORCP estimation process is proportional to the number of ES. Therefore, when the ORCP is estimated using many samples, an equation of a straight line with the same slope can be obtained. Lines with the same slope may cause high computational load and errors because of the innumerable intersections with other lines that appear. Hence, it is necessary to consider the slope in the process of calculating the equation of the straight line. Minimum Difference of Gradient (MDG) is used as the variable to exclude equations of straight lines that have the same slope. When the difference between the slope of the newly calculated line and the slope of the previously calculated line exceeds the MDG, it is used for the ORCP estimation. If the MDG is set to \(\pm 1\) degree, the maximum number of straight lines that can be obtained is 359, even if there are a huge number of samples.  
Figure 8: Distance in xy-plane of ORCP and PCP according to RAE.  
In the verification step, error determination is carried out. TS samples satisfying the HPOM estimated in the hypothesis step are classified as inliers. The quantity of these is acquired as a cost. Error determination is performed by examining the distance on the xy-plane of the ORCP using determination of the object sample and using the RAE of the determined object sample found in Equation (2). However, since the shape of the pupil observed by the camera might be distorted according to the characteristics of the lens, an error in the RAE may occur. Therefore, the Allowable Angle Error (AAE) is reflected by a range variable that sets the allowable error range for the distance in two-dimensional coordinates of the sample being judged. After that, the measured error of the sample being judged is calculated according to Equation (2). This result is then compared and classified as an inlier if it is smaller than the AAE and as an outlier if it is bigger than the AAE. If the cost obtained by error determination is large compared to the maximum cost, samples classified as inliers are stored. This is done so the larger the cost obtained through error determination, the more similar the estimated HPOM is to the actual HPOM. This process is repeated and the optimal HPOM is re-estimated using the inlier obtained when the cost is at its maximum.