## 2 Direct Phase Measuring Deflectometry  
Figure 1 is a schematic diagram of the proposed full-field 3D shape measurement system for specular surfaces using the DPMD technique. The system consists of two LCD screens, one mirror, one CCD camera, and one beam splitter plate. The beam splitter was adjusted to a suitable position that makes a virtual image \(\mathrm{LCD_{1}}^{{}^{\prime}}\) of \(\mathrm{LCD_{1}}\) parallel to \(\mathrm{LCD_{2}}\), as if the two LCD screens were located in two parallel positions. The two screens were parallel to a reference mirror located in the reference plane. The parallelism was established by repeated adjustments of the relative orientations of the three components (the two LCD screens and reference mirror). A checkerboard could be generated and displayed on the \(\mathrm{LCD_{1}}\) and \(\mathrm{LCD_{2}}\) screens, which were both located on the angular displacement platforms. As a result, the screens could change position and direction randomly and accurately. A CCD camera captured the checkerboard images on the two screens. The extrinsic parameters of \(\mathrm{LCD_{1}}\) and \(\mathrm{LCD_{2}}\) were calculated from the captured checkerboard image. Then, based on their differences, the relative orientation between the virtual image of \(\mathrm{LCD_{1}}\) and \(\mathrm{LCD_{2}}\) was adjusted to be parallel, using the angular displacement platforms. This procedure was repeated several times until the three components were nearly parallel. And then there is also [PERSON]'s method [39] for compensation by generating deformed fringes by software to be equivalent to the exact parallel alignment of two display screens. In the diagram, \(h\) is the depth of the measured point with respect to the reference mirror, \(d_{1}\) is the distance between \(\mathrm{LCD_{1}}^{{}^{\prime}}\) and the reference mirror, and \(d_{2}\) is the distance between \(\mathrm{LCD_{2}}\) and the reference mirror. As illustrated in Figure 1, two rays of light were displayed and reflected into the CCD camera via the measured surface and the mirror at the reference position. The two incident rays corresponded to the same reflection light. The absolute phases of the two incident rays are denoted as \(\phi_{1}\) and \(\phi_{2}\) on the reference plane and \(\phi_{1}^{\prime}\) and \(\phi_{2}^{\prime}\) on the measured specular surface. \(\theta\) and \(2\theta+\beta\) are the angle between the incident ray and the normal vector of the reference and the angle between the incident ray and the normal vector of the measured specular surface, respectively. The period of the displayed fringe pattern on the LCD screen is denoted as \(q\). \(l\) is the distance on \(\mathrm{LCD_{1}}^{{}^{\prime}}\) between the two incident rays because of the height and gradient of the measured surface. Parameter \(h\) stands for the height of the measured specular surface with respect to the reference plane. \(d_{1}\) is the distance between \(\mathrm{LCD_{1}}^{{}^{\prime}}\) and the reference mirror, and \(d_{2}\) is the distance between \(\mathrm{LCD_{2}}\) and the reference mirror. According to geometric triangle relationships, which are illustrated in Figure 1, Equations (1)-(4) can be acquired respectively. \[(d_{1}-d_{2})\tan\beta=\frac{(\phi_{1}-\phi_{2})\times q}{2\pi} \tag{1}\]  
\[(d_{1}-d_{2})\tan(2\theta+\beta)=\frac{(\phi_{1}^{\prime}-\phi_{2}^{\prime}) \times q}{2\pi} \tag{2}\]  
Figure 1: Schematic diagram of the measuring system using direct phase measuring deflectometry (DPMD). \[(d_{1}+h)\tan\theta+l=(d_{1}-h)\tan(2\theta+\beta) \tag{3}\] \[l=\frac{(\phi_{1}-\phi_{1}^{\prime})\times q}{2\pi}. \tag{4}\]  
Equation (5) can be derived based on the above Equations (1)-(4).