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Abstract
Deflectometry has been widely used in topography measurement of specular surface. In deflectometry with a curved screen, the range of the gradient and height field of the measured specular surface can be effectively expanded compared to deflectometry with a plane screen. As stereo deflectometry measures gradient with high accuracy, the specular surface is reconstructed by integrating the gradient. In this paper, a stereo deflectometry with a curved screen is proposed in the aspect of system calibration and measuring principle. A pair of cameras and deflectometric system are calibrated simultaneously to obtain the camera parameters and relationship between two cameras and the curved screen. Then, pairs of rectified fringe patterns are demodulated to obtain two pairs of rectified absolute phase maps and homologous points are searched along epipolar line with the principle of the same normal direction on specular surface. Finally, simulated and actual experiments are carried out and the results show high accuracy and stability.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Components with specular surface are widely used in aerospace, automobile, optics system and artificial intelligence. Along with interferometer [1,2] and Hartmann test [3], phase measuring deflectometry (PMD) can efficiently measure gradient and height of specular surface under test (SuT) because of its advantages of full-field data acquisition, non-contact operation, automatic data processing, high accurate data, and large dynamic range [4,5].
Based on the principle of specular surface reconstruction, PMD can be derived into classical PMD [6–8], light-tracking based PMD [9,10], model PMD(MPMD) [11,12], differential-geometry based PMD [13–16], direct PMD(DPMD) [17,18], et.al. Based on the number of imaging device (normally camera) in PMD systems, PMD can be derived into monoscopic PMD (mono-PMD) and stereoscopic PMD (stereo-PMD).
A PMD system usually contains a liquid crystal display (LCD), a computer and a camera in mono-PMD or two cameras in stereo-PMD, as illustrated in Fig. 1. Straight fringe patterns are generated in software and displayed on the LCD. The camera captures the distorted fringe patterns modulated by the measured surface. Phase information is demodulated from the captured fringe patterns to reconstruct three-dimensional (3-D) shape. Conventional PMD reconstructs specular surface by integration of gradient computed from the distorted patterns. In mono-PMD based on light tracking, there are usually a moving LCD [9,10] or two LCD matching a beam splitter [17,18] to orient light and furtherly reconstruct SuT. In mono-PMD based on differential geometry, instead of moving the LCD or adding a LCD, specular surface can be reconstructed by differential property of specular surface. DPMD directly formulates variation of phase and height, instead of gradient of surface with integration, so it is convenient to reconstruct discontinues or isolated specular surface [17,18]. In stereo-PMD, with the principle of consistence of a point’s normal direction on SuT [19], the gradient can be computed. The specular surface can be reconstructed by integration of gradient. In mono-PMD and stereo-PMD, by modeling the specular surface, MPMD solves model parameters and system parameters by iteration.
The calibration of stereo-PMD builds up the relationship among the screen and a pair of cameras. Recently, a calibration method for stereo deflectometric system and cameras through an iterative optimization is studied [20]. To improve the accuracy of calibration, speckle patterns are employed in the calibration method for stereo deflectometry [21]. An on-machine calibration method for stereo deflectometric system is proposed which reduces the number of intermediate parameters and improves the accuracy of calibration [22].
With the gradient computed by stereo-PMD, specular surface can be reconstructed. Traditionally, the zonal reconstruction is carried out along rectangular mesh [23]. However, the obtained gradient data usually locate on a quadrilateral geometry in the camera’s coordinate, instead of a rectangular mesh. To solve this problem, reconstruction methods on the quadrilateral geometry are proposed, such as high-order least-squares method [24], radial basis functions method [25] and the method in general form of zonal reconstruction [26], et. al.
When a plane screen employed in PMD, because of screen’s shape and size, some part of SuT will reflect scene outside the plane screen, resulting that SuT can’t be measured in single measurement. As a large-curvature specular surface is placed in front of a curved screen, a curved screen can effectively solve this problem because of its distinguishing characteristic. The effect of a curved screen in expanding measurement range of specular surface is discussed elaborately in [27]. Therefore, a curved screen with small size can measure larger specular surface than a plane screen. The measuring principle and the calibration methods of mono-PMD with the curved screen have been studied [27]. With excellent performance of improving the accuracy and the measurable range, a stereo-PMD with a curved screen has been studied in [28].
Different from the stereo-PMD system in [28], this paper proposed a stereo-PMD with a curved screen to measure large-curvature specular surface in single measurement. In calibration of system, this paper calibrates a pair of cameras and system simultaneously by minimizing reprojection error in a procedure of iteration, instead of separated calibration of every camera and the curved screen in [28]. In the principle of measurement, this paper reconstructs SuT by integration of gradient in common field of view of the pair of cameras, instead of registration of point cloud from the pair of cameras in [28]. In the proposed stereo-PMD, combined with solved gradient of SuT, there will be an initial solved 3-D coordinate of SuT. In PMD systems, the solved gradient tends to be more accurate than the solved z component of initial 3-D coordinate. To the extent that with an absolute uncertainty of z component up to 0.1mm, the absolute error of slope is less than 100 arcsec [29]. With the solved gradient and x, y components of the initial 3-D coordinate, SuT can be reconstructed more accurately by integration.
In the following, Section 2 explains the system calibration and measuring principle of the proposed stereo-PMD with a curved screen. In Section 3, the simulated and actual experiments have been performed to verify the proposed stereo-PMD. Section 4 gives conclusion and future directions.
2. Principle and calibration
This section presents the principle and calibration of the stereo-PMD with a curved screen.
2.1 Stereo-PMD with a curved screen
A diagram of the proposed stereo-PMD system is shown in Fig. 2. It contains two cameras and a curved screen. A pair of cameras are mounted on the top of the curved screen. After calibration of the stereo-PMD system, a SuT is measured. The screen displays phase-shifted [30] horizontal and vertical sinusoidal fringes with fringe numbers N2, N2-1, N2-N selected by the optimum fringe frequency method [31,32]. The displayed fringes are reflected by SuT, and then the distorted fringes are captured by the two cameras. In configuration of the system, SuT and the pair of cameras should be close to the middle of the curved screen in the horizonal direction. With a certain curved screen, the distance between SuT and the curved screen play a major role in extending range of measurement. Thus, on the premise that screen does not shield SuT from the perspective of the pair of cameras, SuT should be as close as possible to the curved screen.
In consideration of efficiency, the recorded fringe patterns are rectified by the results of calibration. The rectified image of a point in measurement field locates on the same row of images pair which will accelerate matching of homologous points. After rectification, the rectified fringe patterns are demodulated to obtain two pairs of absolute phase map in the horizontal and vertical direction. However, specular surface reflects fringe patterns, resulting in different phase values in the pair of cameras.
Every pixel in the pair of cameras corresponds to a position on the screen which can be obtained by interpolation of the absolute phase. In the rectified phase map pairs, homologous points locate on the same row of images. Once a base pixel in camera1 is chosen, searching of the homologous pixel is along the row in camera 2. Along the row, every pixel in camera 2 orients a point by triangulation with the base pixel. If the point is on SuT, the bisectors of light corresponding to camera 1 and 2 are the closest. With this principle, the homologous point of pixel level in camera 2 can be searched. The homologous point in subpixel level can be searched by the interpolated phase value in camera 2. The detailed procedure of searching is illustrated in Fig. 3.
With a base point q on camera 1, it captures screen pixel of the reflected light A. After computing difference of bisectors of camera 1 and 2 along base pixel q’s row, p2’s bisector is the closest to q’s corresponding to D2. p1 and p3 corresponding to D1 and D3 are adjacent pixels to p2. B1, B2 and B3 has the same phase value to p1, p2 and p3, respectively. In the searching range between p3 and p1, a subpixel p’s phase value and the corresponding position in screen can be obtained by interpolation among p3, p2 and p1. As bisectors of camera 1 and camera 2 are denoted as n1 and n2 respectively, searching range is decreased by an iteration procedure, as illustrated in Fig. 4. When the distance between p1 and p3 is less than ${{\boldsymbol \varepsilon }_\textrm{p}}$ and the cross product between n1 and n2 is less than ${{\boldsymbol \varepsilon }_\textrm{n}}$, the iteration terminates with solving p = p2. A subpixel p corresponding to D can be obtained with satisfied similarity of bisectors. With triangulation, 3-D coordinate of point D can be obtained. The bisector of $\angle \textrm{ }{\textrm{F}_\textrm{1}}\textrm{DA}$ is normal direction of SuT and the corresponding gradient of point D can be computed.
Although initial 3-D coordinate of SuT is obtained, the gradient is more accurate than the 3-D coordinate. Combining the gradient, x and y components are extracted to reconstruct specular surface by integration. In stereo-PMD system, two cameras merely record the distorted fringe patterns with the screen at a fixed position. However, in mono-PMD with a curved screen [27], a camera needs to record both normal and distorted fringe patterns with different positions of the screen which is one of the main sources of error. Thus, the stereo-PMD tends to be more accurate than a mono-PMD.
2.2 Calibration of stereo-PMD with a curved screen
The calibration of stereo-PMD with the curved screen is to confirm relationship among the pair of cameras and the curved screen. Because it has a changeable radius, the commercial curved screen is reconstructed and a virtual plane with height h and width w is fixed on the curved screen by displaying featured points. By minimizing reprojection error of the featured points to the pair of cameras, the stereo-PMD calibration method solves intrinsic parameters of cameras and extrinsic parameters among the pair of cameras and the virtual plane.
A flat mirror is used to reflect the displayed pattern. Two cameras record n reflected patterns corresponding to n postures of the flat mirror. In every recorded picture, m featured points will be extracted. As the camera 1 is selected as a base camera, transformation of camera 2 to the curved screen and the flat mirror can be computed by a rotation matrix Rstereo and a translation vector Tstereo between camera 1 and camera 2. With the recorded patterns, initial value of Rstereo and Tstereo can be computed by the method in [33]. A point Pleft in camera 1’s coordinate can be transferred to Pright in camera 2’s coordinate by the following equation.
The diagram of calibration is given in Fig. 5. The initial values of rotation matrix Rvp and translation vector Tvp between camera 1 and the virtual plane can be computed by the method in [34,35]. At a posture i(i = 1, …, n) of the flat mirror, vdi denotes a vector in camera 1’s coordinate from the mirror to the optical center of camera 1 along the mirror’s normal direction nmirror. The coordinate of a featured point Pchecker on the virtual plane is X in the checker coordinate system and its virtual image’s coordinate in camera 1 coordinate system and camera 2 coordinate system are denoted as pleft and pright, respectively. According to relationship of mirror symmetry between a real image and its virtual image, Eq. (2) transforms X to pleft with geometry parameters of the flat mirror Rvp, Tvp and vdi. As initial relationship Rstereo and Tstereo between two cameras have been calibrated, the virtual image of X can be transformed to pright in camera 2’s coordinate by Eq. (3).
The nonlinear mathematic model of the left camera and the right camera is marked as ${\Re _{\textrm{left}}}$ and ${\Re _{\textrm{right}}}$, respectively. The intrinsic parameters of the left camera and the right camera are marked as ${A_{\textrm{left}}}$ and ${A_{\textrm{right}}}$, respectively. A cost function is expressed as the following equation to compute distance between the reprojected featured points and the recorded featured point ${{\boldsymbol p}_{_{\textrm{left\_}i{\_}j}}}^{\prime}$, ${{\boldsymbol p}_{_{\textrm{right}\_i{\_}j}}}^{\prime}$. The calibration results are obtained to minimize the cost function (4) by the Levenberg-Marquardt algorithm.
3. Experiment
3.1 Simulated experiment
For testing the proposed stereo-PMD with a curved screen, some simulation experiments have been carried out. A curved screen and a pair of cameras are placed as shown in Fig. 2. A right-handed world coordinate’s xoy plane is fixed on the reference plane. The curved screen is a part of cylinder with a radius of 1000 mm whose axis is parallel to y axis of the world coordinate and passes through (0, 0, 800). The rotation matrix and the translating vector between camera 1 and the reference plane are $\left( \begin{array}{l} \textrm{ 0}\textrm{.9976 0 0}\textrm{.0698}\\ \textrm{ 0}\textrm{.0061 0}\textrm{.9962 - 0}\textrm{.0869}\\ \textrm{ - 0}\textrm{.0695 0}\textrm{.0872 0}\textrm{.9938} \end{array} \right)$ and $\left( \begin{array}{l} \textrm{ }0\\ \textrm{ }0\\ 402.5084 \end{array} \right)$, respectively. A point Pcamera1 in camera 1 can be transferred to Pcamera2 in camera 2 by a rotation matrix $\left( \begin{array}{l} \textrm{ 0}\textrm{.9945 0}\textrm{.0091 - 0}\textrm{.1041}\\ \textrm{ - 0}\textrm{.0127 0}\textrm{.9993 - 0}\textrm{.0342}\\ \textrm{ 0}\textrm{.1037 0}\textrm{.0354 0}\textrm{.9940} \end{array} \right)$ and a translating vector $\left( \begin{array}{l} \textrm{ 41}\textrm{.9135}\\ \textrm{ 13}\textrm{.7796}\\ \textrm{ 13}\textrm{.2478} \end{array} \right)$. The diagram of simulated system in world coordinate is illustrated in Fig. 6. A specular cylinder with a radio of 60 mm has been measured.
Noises with different levels were added into the generated fringe patterns by
where Inoisy denotes the generated noisy fringe pattern, i is the noise level and i = 0, …, 10, randn is a matrix whose element obeys standard normal distribution.At different noise levels, the RMS (Root Mean Square) distance between the integrated height and the real height is illustrated in Fig. 7.
The simulated experimental results indicate that the proposed stereo-PMD is of high noise resistance and high reconstruction accuracy.
3.2 Stereo-PMD with a curved screen
3.2.1 Hardware of the stereo-PMD system
The stereo-PMD system consists of a curved screen, two CCD cameras, and a translating stage, as depicted in Fig. 8. The curved screen is fixed on the translating stage. The cameras (XIMEADE XIQ, MQ042CG-CM, Germany) have a resolution of 2048 × 2048 pixels, and use a standard prime lens (AZURE-1620ML5M) of focal length 16 mm. The model of the curved screen is SAMSUNG C24T55* 1000R, with a resolution of 1920*1080, pixel size of 0.272 mm*0.272 mm. The translating stage (DHC, GCD-20Series, Beijing, China) has an accuracy of 1 µm.
3.2.2 System calibration
Calibration of the stereo-PMD is to confirm relationship among the pair of cameras and the curved screen. The curved screen was reconstructed by the method in [27]. Two cameras were placed in front of the curved screen with the displayed fringe patterns. In order to reconstruct the curved screen, pixels on phase maps of two cameras with the same phase value were treated as homologous point. With triangulation between two cameras, the depth of homologous point is computed. Combined with phase value of every homologous point, the point cloud of the curved screen is saved. In order to represent the curved screen, a virtual plane was fixed on the screen by displaying checker pattern at the stage of screen reconstruction, as illustrated in Fig. 9.
After reconstructing the curved screen, a pair of cameras and the curved screen were placed as shown in Fig. 2. A flat mirror was used to reflect the displayed checker pattern. With thirty different postures and positions of the flat mirror, thirty reflected checker patterns were recorded and featured points were extracted. With the recorded checker patterns, two cameras were calibrated to obtain initial calibration results, such as intrinsic parameters and transformation parameters among two cameras and vdi of camera 1. Then, the initial calibration results were substituted into Eq. (4). After iteration, the calibration results are computed by the distribution of reprojection error, as shown in Fig. 10 and the lowest reprojection errors are [0.0638 0.0485] pixels.
As relationship among the pair of cameras and the curved screen has been calibrated, the model of the curved screen was transferred to camera 1. The calibrated stereo-PMD system in camera 1 is given, as illustrated in Fig. 11, where color bar represents depth of the curved screen in camera 1 coordinate system.
3.2.3 Experiments on a specular surface
In order to evaluate the proposed stereo-PMD method, the system measured a specular cylinder and a specular step.
The specular cylinder is shown in Fig. 12(a). The pair of cameras recorded the reflected fringe patterns, as shown in Fig. 12(b)-(c). After rectification of image pairs, two pairs of absolute phase map in the horizonal and vertical direction were demodulated, as shown in Fig. 12(d)-(e). In order to evaluate the reconstruction by the proposed stereo system, the method reconstructing specular surface in [13–15] was treated as a benchmark to a quantitative measurement. Partial differential equations of depth along x and y direction of coordinate of camera 1 were set up. By computing difference of integral depth with different integration sequence in a mask, the depth is final reconstruction which corresponds to the minimum difference of integral depth, as illustrated in Fig. 13(a). The homologous pixel was searched by the method in Section 2.3. Initial reconstruction of specular cylinder by the stereo-PMD system was obtained, as shown in Fig. 13(b). After integration of gradient, the specular cylinder was reconstructed, as shown in Fig. 13(c). The distance of initial reconstruction and integral reconstruction by the stereo-PMD system to the benchmark is shown in Fig. 14(a) and Fig. 14(b), with RMS distance 0.1872 mm and 0.0035 mm respectively. The results indicate that the integral reconstruction is more accurate than the initial reconstruction in the stereo-PMD system with a curved screen.
Fig. 12. Specular cylinder, reflected fringe patterns and rectified phase maps. (a) specular cylinder, (b)-(c) reflected fringe patterns, (d)-(e) rectified phase maps.
Fig. 13. Reconstruction of specular cylinder. (a). Reconstruction by method in [13–15], (b). initial reconstruction by stereo-PMD, (c). reconstruction by stereo-PMD with integration.
Fig. 14. Error of reconstruction to the benchmark. (a). initial reconstruction by stereo-PMD, (b). reconstruction by stereo-PMD with integration.
The specular step is shown in Fig. 15(a). The pair of cameras recorded the reflected fringe patterns, as shown in Fig. 15(b)-(c). Two pairs of the rectified absolute phase map in the horizonal and vertical directions were demodulated, as shown in Fig. 15(d)-(e). The homologous pixel was searched by the method in Section 2.3. Initial reconstruction of specular step by the stereo-PMD system was obtained, as shown in Fig. 16(a). After integration of gradient, every isolated surface was reconstructed separately and then transferred a mean distance between initial reconstruction of the surface and integral reconstruction. After transferring every isolated surface, the specular step was reconstructed, as shown in Fig. 16(b).
Fig. 15. Specular step, reflected fringe patterns and rectified phase maps. (a) specular step, (b)-(c) reflected fringe patterns, (d)-(e) rectified phase maps.
Fig. 16. Reconstruction of specular step. (a). Initial reconstruction by stereo-PMD, (b). reconstruction by stereo-PMD with integration.
The actual distance between steps was measured by a C32-bit ZEISS Calypso coordinate measuring machine. The measured coordinates of every step were fitted to a plane. The distance between the points of the adjacent step and the fitting plane was treated as measured distance. Table 1 shows the comparison between the actual distance and the measured distance.
Table 1. Actual distance, measurement distance of step and error by three calibration methods (mm)
The results indicate that reconstruction of the specular surface by the proposed stereo-PMD system after integration is more accurate than the initial result. Furthermore, the proposed stereo-PMD with a curved screen reconstructs specular surface with higher accuracy than the mono-PMD system with a curved screen in [27]. Moreover, the RMS distance between integral reconstruction and the fitting plane in a range between 10−3 and 10−5 mm is much lower than the mono-PMD system with a curved screen and initial reconstruction in the proposed stereo-PMD.
4. Conclusion
In this paper, a stereo-PMD with a curved screen has been proposed to measure specular surface. The gradient of specular surface is measured with the principle of normal direction consistence. The light direction corresponding to every camera pixel is oriented by interpolation of the phase value on the screen model. The homologous pixels in subpixel level can be searched by an iteration procedure. Simulated and actual experiments show high accuracy and stability of the proposed stereo-PMD with a curved screen. In the future, further experiments will be carried out to measure more components with specular surface.
Funding
National Natural Science Foundation of China (51675160, 52075147).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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