## 4 Discussion
### Difference and Advantage  
The P3P is the minimal subset of PnP and in some cases, it is difficult to acquire 3D points. This is the reason why we do our work with the P3P problem. However, P3P solvers needed a fully calibrated camera in previous studies, so we had the idea about extrinsic parameter estimation with an uncalibrated camera and three 3D points. From Figure 1, although \(\alpha\), \(\beta\), \(\gamma\) cannot be computed with an uncalibrated camera in the original camera frame, they can be computed in the original world frame when the camera position is known. This finding let us convert our problem to the known P3P problem as shown in Figure 3. The thought is to build a new camera system whose camera frame is defined with the original world frame and the world frame is defined with the original camera frame. This is a P3P problem between the new camera frame and the new world frame. With the definition of a new camera system, three new 2D-3D point correspondences are obtained and then the extrinsic parameters of the new camera are obtained by the P3P solvers. There is an interesting point in this solution--the camera position in the new world frame contains the intrinsic parameter information of the original camera system, as shown in Figure 4. This is the key for intrinsic parameter estimation in our proposed method.  
Additionally, the way to solve the multi-solution problem is different. The existing P3P solvers use a fourth point to disambiguate multi-solution phenomena, but our proposed method uses no extra point. The multi-solution phenomenon of P3P solvers in the new camera system leads to the multi-solution of the intrinsic parameter in the original camera system and hence we can obtain up to four principal points. This is because the image resolution is known and the ideal principal point can be obtained with Equation (14). The ideal principal point, not a fourth 3D point, can be used to disambiguate multi-solution phenomenon in our proposed method. In this paper, we assume the one closest to the ideal principal point is the unique principal point and this assumption yields good results with synthetic data and real images, as shown in Section 3.  
In the new camera system, the new 2D image points are obtained from original 3D points which have the characteristics of high precision because they are measured by total station or other high precision measurement equipment. Therefore, the error of the new 2D point can be an order of magnitude lower than the error of the original 2D point and we think this may be a reason why our proposed method has improved numerical stability and lower noise sensitivity. Compared to the P3P method, only the establishment of the new frames and coordinate transformation are added in our proposed method, which can be made very fast, and this is the reason why our proposed method is just slightly slower than [PERSON]'s method. Our proposed method just uses three 3D points and this is a reason why it is faster than the GP4 Pf that uses four 3D points.  
In brief, we can see that the proposed method has the following advantages--(1) it just uses three 3D points and the camera position to estimate intrinsic and extrinsic parameters; (2) it has no multi-solution phenomenon; and (3) it has improved numerical stability and lower noise sensitivity.