## 2 Materials and Methods
### Problem Statement and Method  
#### 2.3.1 Problem Statement  
The strict requirement of full calibration for the camera in the P3P solvers prevents its application from these cases where some intrinsic parameters might change online or be unknown. In this paper, these intrinsic parameters are the focal length, principal point and the problem is to solve the intrinsic and extrinsic parameter estimation by solving the P3P problem with a known camera position.  
When the focal length and principal point are unknown, the \(\alpha\), \(\beta\), \(\gamma\) cannot be computed in the P3P problem. Some algorithms are proposed for extrinsic parameter and focal length  
Figure 1: The P3P problem.  
estimation by four 3D points [21] or for extrinsic parameter, focal length and principal point estimation by five 3D points [25]. However, user convenience dictates the use of as few 3D points as possible--accurate 3D points are troublesome and expensive to acquire and maintain. In some cases where the camera is fixed, the camera position can be obtained as prior knowledge and then our problem is to solve intrinsic and extrinsic parameter estimation with fewer 3D points and the known camera position. The problem is illustrated in Figure 2--let \(O_{C}\) be the center of perspective which is known, \(X_{i}(i=1,2,3)\) be the 3D reference points, \(x_{i}(i=1,2,3)\) be their image projections and the task is to solve intrinsic (the focal length, principal point) and extrinsic parameter estimation from these information.  
In Figure 2, \(\alpha,\beta,\gamma\) are computed by three 3D points \(X_{i}(i=1,2,3)\) and the camera position \(O_{C}\), not by camera intrinsic parameters (focal length, principal point) and image projections \(x_{i}\) as in the traditional P3P problem as shown in Figure 1. Now our problem is summarized geometrically as follows--the image points are \(x_{i}(i=1,2,3)\) in the original image frame and let \(|x_{1}x_{2}|=a\), \(|x_{2}x_{3}|=b\), \(|x_{3}x_{1}|=c\); we assume \(|O_{c}x_{1}|=d_{1}\), \(|O_{c}x_{2}|=d_{2}\), \(|O_{c}x_{3}|=d_{3}\) and from triangles \(O_{c}x_{1}x_{2}\), \(O_{c}x_{2}x_{3}\), \(O_{c}x_{3}x_{1}\), we obtain the equations:  
\[\begin{array}{l}d_{2}^{2}+d_{3}^{2}-2d_{2}d_{3}\cos\alpha=a^{2}\\ d_{3}^{2}+d_{1}^{2}-2d_{3}d_{1}\cos\beta=b^{2}\\ d_{1}^{2}+d_{2}^{2}-2d_{1}d_{2}\cos\gamma=c^{2}\end{array}\]  
Now the task is to find a set of solutions for \(d_{1},d_{2},d_{3}\) which is known to provide up to four possible solutions. This is similar to the P3P problem, but the variables and invariants of the equations have completely different meanings--\(a\), \(b\), \(c\) are the distances between every two image points, but they are the distances between every two 3D points in the P3P problem; \(d_{i}\) is the distance between camera position \(O_{C}\) and image point \(x_{i}(i_{i},\upsilon_{i})\), but is the distance between the camera position \(O_{C}\) and 3D point \(X_{i}\); in addition, the way to obtain the values of \(\alpha,\beta,\gamma\) is different, as described above.  
In the following, our main work is to convert our problem to a P3P problem and obtain the values of intermediate variables which are then used for intrinsic and extrinsic parameter estimation and to lastly find the unique solution by image resolution. We provide our method of the problem as follows.