## 2 Principle
### Methods  
#### 2.2.1 Entrance Pupil Calibration Method (EPCM)  
In Figure 1a, because both the entrance pupil center and the external stop lie on the optical axis in the paraxial region of the camera lens, a reference surface and an external stop are used to obtain the entrance pupil center, which is usually a virtual image of the aperture stop of the camera lens and is inside the lens. The origin of the global coordinatesystem \(O-XYZ\) is established as the point \(O\), which is the coordinate of the external stop on the camera lens. Additionally, point \(E\) is the entrance pupil center.  
Based on the geometry shown in Figure 1a, where the points \(E\) and \(O\) are along with the optical axis, from the view of ray tracing in reverse we extend the ray of \(EO\) to the reflection point \(M_{r0}(x_{m0},\,y_{m0})\) on the reference surface and reach spot \(S\) on the LCD screen. These four points, \(O\), \(E\), \(M_{r0}\), and \(S\), are along the optical axis. Assuming that the reference surface is an ideal plane, according to the law of reflection, the incident angle is equal to the reflection angle. The reflection points of the reference surface can be obtained, as shown in Equation (2):  
\[\begin{array}{l}x_{mr}=\frac{x_{sr}z_{m0}+x_{s}z_{m0}}{z_{m0}+z_{m0}}\\ y_{mr}=\frac{y_{sr}z_{m0}+y_{m0}z_{m0}}{z_{m0}+z_{m0}z_{m0}}\end{array}, \tag{2}\]  
where \(z_{m2s}\) and \(z_{m20}\) are the distance between the reference surface and the screen and the distance between the reference surface and point \(O\) in the \(z\) direction, respectively; \(x_{sr}\) and \(y_{sr}\) are the coordinates of the screen pixels corresponding to the central area of the image plane (as shown in Figure 1b); \(x_{o}\) and \(y_{o}\) are the coordinates of point \(O\), which can  
Figure 1: (**a**) Schematic diagram of the calibration of the entrance pupil center (point \(O\) is the external stop, point \(E\) is the entrance pupil center, and point \(M_{r0}\) is a reflection point on the reference surface); (**b**) illustration of central area of the image.  
be measured by the coordinate measuring machine (CMM). Significantly, point \(M_{r0}\) is one of the reflection points and is on the optical axis.  
For convenience, the auxiliary points, \(M_{r1}\), \(M_{r2}\), \(B_{1}\), and \(B_{2}\), are depicted in Figure 1a. They are perpendicular to the \(x\)-axis and the \(y\)-axis, respectively. \(M_{r1}\) and \(M_{r2}\) are equal to point \(O\) and point \(E\) in the \(Y\) direction, respectively. Similarly, \(B_{1}\) and \(B_{2}\) are equal to point \(O\) and point \(E\) in the \(Y\) direction. As shown in Figure 1a, the angle relationships between the camera and the reference surface can be expressed in triangles \(M_{r1}OB_{1}\) and \(M_{r0}OM_{r1}\), as shown in Equation (3):  
\[\begin{array}{l}\alpha_{1}=\arctan\left(\frac{x_{m\
u 0}-x_{o}}{z_{m 2o}}\right)\\ \beta_{1}=\arctan\left(\frac{y_{m\
u 0}-y_{o}}{d_{m2o}}\right)\end{array}, \tag{3}\]  
where \(d_{m2o}=\sqrt{\left(x_{m\
u 0}-x_{o}\right)^{2}+\left(y_{m\
u 0}-y_{o} \right)^{2}+\left(z_{m2o}\right)^{2}}\).  
Based on triangles \(M_{r0}OM_{r1}\) and \(M_{r0}EM_{r2}\), triangles \(M_{r1}OB_{1}\) and \(M_{r2}EB_{2}\) are similar to each other. We assume that the distance between point \(O\) and point \(E\) is \(d_{p}\). According to the above, the coordinates of the entrance pupil center can be determined in the global coordinate system, as shown in Equation (4):  
\[\begin{array}{l}d_{m2e}=d_{m2o}+d_{p}\\ z_{m2e}=d_{m2o}\cos(\beta_{2})\cos(\alpha_{2})\\ x_{e}=x_{m\
u 0}-z_{m2e}\tan(\alpha_{2})\\ y_{e}=y_{m\
u 0}-d_{m2e}\sin(\beta_{2})\\ z_{e}=z_{m2e}-z_{m2o}+z_{o}\end{array}, \tag{4}\]  
where \(x_{e}\), \(y_{e}\), and \(z_{e}\) are the coordinates of the entrance pupil center, which are the coordinates of the camera, \(\alpha_{2}=\alpha_{1}\) and \(\beta_{2}=\beta_{1}\). After obtaining the camera coordinates, the reflection points coordinates can be solved using an iterative reconstruction strategy [28].