## 1 Introduction  
Phase measuring deflectometry (PMD) with structured light projection and the phase-shifting technique is a highly accurate optical surface measuring method based on the law of light reflection. It has the advantages of high speed, low cost, and a high dynamic range; furthermore, it requires a simple device and is non-destructive [1; 2; 3; 4; 5; 6; 7; 8; 9; 10]. In the PMD system, rays are considered to be emitted from the projection center of a camera lens and reflected off a planar surface under test (SUT) to reach a liquid crystal display (LCD) screen. During the measurement process, the calibration step is carried out to establish the relationships among the screen coordinate system, the coordinate system of the SUT, and the camera coordinate system; then, the slope of the reflection points on the SUT are obtained according to the slope calculation equation [9; 10]. Finally, the slope data are integrated to obtain the surface information [3; 4; 5; 6; 7; 8; 9; 10]. Among these variables, the most important are the coordinate positions of the screen, the SUT and the projection center of the camera. The calibration error in them will degrade the measurement accuracy of the SUT.  
At present, an external pinhole is utilized as the projection center of the camera, whose position can be accurately measured with a laser tracker in PMD or the Software Configurable Optical Test System (SCOTS) [3; 11; 12]. The projection center of the camera lens can also be calibrated based on the photogrammetry principle [13; 14]. Based on [PERSON]'s technique, [PERSON] et al. proposed a calibration method with a flat mirror to obtain the position of the projection center of the camera [15; 16]. Additionally, [PERSON] et al. proposed a method to calculate the position of the projection center in a unified world coordinate system from the external parameters (including the translation vector and rotation matrix)of the camera [16; 17]. [PERSON]'s technique requires taking several images from different orientations and different poses, but it is not always accessible in deflectometry, especially for a machine-integrated system.  
Optically, the chief rays reflected off the SUT will certainly converge at the entrance pupil center of the camera lens [18; 19; 20; 21]. The role of the projection center in the PMD system, as is mentioned in the references [13; 14; 15; 16; 17], is equivalent to this center. For the sake of distinction, we named it the entrance pupil center in our method, but it is still called the projection center in other methods.  
For an actual lens, the position of the entrance pupil center can vary as the off-axis angle of an incoming ray increases [18]. However, for most of the lens, this effect is not significant, and thus it can usually be ignored in camera calibration [19]. Therefore, a method based on the calibration of the entrance pupil center is introduced in this paper. The entrance pupil calibration method (EPCM) that we propose is compared with the popular photogrammetric method based on [PERSON]'s technique (PM) [16], and [PERSON]'s MPMD method (MPMD) [7]. A planar window glass about 90 mm in diameter and an optical planar element approximately 103 mm in diameter are measured separately by the three methods, and the measurement results are analyzed.  
The rest of the paper is organized as follows: Section 2 illustrates the basic principle of the three methods (EPCM, PM, and MPMD). Section 3 describes the measurement procedures of the three methods. Section 4 presents the calculation of the perturbations of the degrees of freedom in the PMD system. Section 5 shows the experimental results, including the comparison of the camera calibration and measurement accuracy. Section 6 discusses and summarizes this paper.