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Abstract
Deflectometry has been widely used in topography measurement of the specular surface due to the merits of full-field data acquisition, automatic data processing, high accuracy, and large dynamic range. Existing deflectometry-based methods, such as model phase measuring deflectometry, direct phase measuring deflectometry, and stereo deflectometry, employ a plane screen, so they have a small field view. The characteristics of curved screens can effectively expand the range of the gradient and height field of the measured specular surface. This paper presents a novel deflectometric technique by using a curved screen to measure large-curvature specular surface. In order to obtain a model of the curved screen, a stereo vision method has been proposed to capture fringes displayed on the curved screen. After phase correction along epipolar lines and homologous points matching, the point cloud containing location and phase is obtained. A virtual plane fixed on the screen is assumed to gain posture and position of the screen. The model of the curved screen is transferred to the deflectometric system through the virtual plane. After calibrating the deflectometric system, height of specular objects can be obtained. Simulated and actual experiments on measuring specular objects have been carried out and the results show high accuracy and large field view for large-curvature specular surface of the proposed technique.
© 2021 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. Phase measuring deflectometry (PMD) have been widely used to 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 [1].
A PMD system usually contains a display screen, an imaging device (normally a camera) and a computer. Straight fringe patterns are generated in software and displayed on the screen. The camera captures the distorted fringe patterns modulated by the measured specular surface. Phase information is demodulated from the captured fringe patterns to reconstruct three-dimensional (3D) shape, as illustrated in Fig. 1.
One challenging problem in PMD is the ambiguity of height and gradient, which means the same captured phase matches different height-gradient conditions [2]. In order to solve the ambiguity of height and gradient, many methods have been studied, such as stereo PMD [3,4], model PMD (MPMD) [5–7], different grating planes as phase displayers [8], a moving diffusive structured light source combined with light tracking technique [9,10], reconstruction from single distorted image combined with mathematic analysis [11,12].
In traditional PMD-based methods, the relationship between phase and slope of the measured surface is built, so that slope integration procedure is employed to compute height of the specular surface [13]. However, these methods are difficult to measure discontinues or isolated specular surfaces because of the limitation of integration procedure. A direct PMD (DPMD) [14,15] method has been presented to measure discontinues or isolated specular surface. DPMD directly formulates height and variation of phase, instead of gradient of surface with integration.
A plane liquid crystal display (LCD) screen or plane gratings are employed to generate sinusoidal structured light in all existing deflectometry systems. Due to the size and shape of the used plane LCD screen, some part of the measured surface will reflect scenes outside of screen, resulting in incomplete reconstruction of the specular surface. Thus, specular surface with large curvature can’t be obtained in a single measurement. Therefore, it is necessary to study a new method to measure 3D shape of specular surface with large curvature. The characteristics of curved screens can effectively expand the range of the gradient and height field of the measured specular surface, as illustrated in Fig. 2. Therefore, in this paper, a novel deflectometric technique has been presented to measure large-curvature specular surface by using a curved screen.
Fig. 2. Comparison of measuring field view and virtual image by using plane screen and curved screen.
In the following, Section 2 explains the advantages, measurement principle, calibration method, and model reconstruction of the curved screen. In Section 3, simulated and actual experiments have been performed to verify the proposed deflectometric technique. Section 4 gives conclusion and future directions.
2. Principle
This section firstly presents the principle in PMD system with a curved screen. Then, the relationship between phase and height, and calibration are demonstrated. In order to get a model of the curved screen, a stereo vision method will be studied. Finally, usage of the model is explained.
2.1 Curved screen in deflectometry
In fact, a curved screen can effectively expand the range of gradient and height field of SuT, as illustrated in Fig. 2. When a large-curvature specular surface is measured, the small-size curved screen can satisfy with the measuring field view, instead of a large-size plane screen.
2.2 Deflectometry using a curved screen
A diagram of the proposed deflectometric technique is shown in Fig. 3. It contains a CCD camera and a curved screen with known radius. Ideally, the curved screen is a part of cylinder and radius is an important parameter with a constant value. The CCD camera is mounted on the top of the curved screen. The screen displays horizontal and vertical sinusoidal fringes with fringe numbers N2, N2-1, N2-N selected by the optimum fringe frequency method [16,17]. The displayed fringes are reflected by the reference plane mirror or SuT, and then be captured by the camera. In order to orient light direction corresponding to every camera pixel, the screen is fixed on a precise translating stage to locate two positions. The camera captures the fringes reflected by SuT and the reference plane mirror at the two screen positions.
In order to simplify the geometric relationship, top view and left view of the deflectometric technique in Fig. 3 are illustrated in Fig. 4 when the curved screen is located in two positions. I1 and I2 are the middle point of the curved screen. For a certain pixel in CCD, A1, A2 are the corresponding pixels in the screen reflected by the reference plane, and the same for B1, B2 reflected by SuT. System parameters of the camera’s optical center F(xF, yF, zF), radius of the curved screen Rscreen, I1O and I2O can be predetermined. The relationship between height and systematic parameters can be easily derived in the following.
Every pixel in the camera corresponds to four positions on the curved screen, A1, A2, reflected by the reference plane and B1, B2 reflected by SuT. These four points’ coordinates can be calculated by
${k_{\textrm{A1A2}}}$ and ${k_{\textrm{B1B2}}}$ represent direction of the light A1A2, B1B2 in the left view, as illustrated in Fig. 4(b). ${k_{\textrm{A1A2}}}$, ${k_{\textrm{B1B2}}}$ can be calculated by
In y direction, the distance between optical center F and B1 is expressed by
2.3 Model reconstruction of a curved screen
The derivations in section 2.2 are based on the assumption of the curved screen having a constant radius of curvature. However, the actual commercial curved screen is designed for human viewing and the radius of curvature has a changeable value.
In order to reconstruct a model of the actual screen at two locations, a stereo vision method is studied [18–20]. The screen displays horizontal and vertical sinusoidal fringes selected by the optimum fringe number method and a multiple-step phase-shifting algorithm [16,17], distinguishing screen pixels’ coordinates along the screen and benefiting searching of the homologous points. Every kind of fringe shifts n times with a step of $\mathrm{2\ast }\pi /n$. Two cameras are fixed in front of the screen. After rectifying and analyzing the character of image pair, phase values along the epipolar line are fitted and corrected by a high-degree polynomial.
After demodulating the captured fringe patterns, horizontal and vertical absolute phase maps corresponding to every point location at two screen positions are obtained. The curved screen displaying the sinusoidal fringe patterns is a continuous curved surface, and the intersection line between the epipolar plane and the screen is a continuous curve, as shown in Fig. 5. The phase value on this curve is also continuous, without step phenomenon. After rectification, a camera pinhole model projects the curve to the same row of stereo image plane. Therefore, the function having different horizontal pixel coordinates with the same vertical coordinates as arguments and corresponding phase values as dependent are also continuous. The phase values captured by the two cameras are related to external parameters ${R_{CS}}$, ${T_{CS}}$ between the camera and the screen, and internal parameters ${P_{intrinsic}}$ of the camera. As above three parameters shown, shape of the screen determines the captured phase values. In other words, due to different shape of the screen, the same pixel of the camera will capture different phase values. Denoting hi as position of the screen point corresponding to a pixel of the camera, i as the horizontal camera pixels coordinate, f as the function of above parameters, the captured phase value phasei can be expressed as
Through the N-order polynomial fitting of discrete points, a distribution function of phase values on the camera epipolar line can be determined by
Due to the influence of electronic devices’ noise, the fringe pattern has non-negligible noise in the actual measurement. It’s effective to resist noise by fitting phase along the epipolar line in the image pair. Based on the fitting function, the matching points’ horizontal coordinates corresponding to the base points’ phase values can be computed, and the difference of horizontal coordinates of base and matching points is the disparity value. Depth of screen can be computed by using the obtained disparity value.
2.4 Calibration of the deflectometric system
In the proposed deflectometric technique based on the curved screen, xoy plane of the right-hand system coordinate is fixed on the reference mirror. Optical center F and coordinates of the screen model need to be calculated in system coordinate according to Eqs. (5) and (6).
Optical center F can be determined by capturing the markers with known separation on the reference mirror. Meanwhile, rotation matrix ${R_{mirror}}$ and translation vector ${T_{mirror}}$ between the camera and the reference mirror, normal direction of the reference mirror ${n_{mirror}}$ in camera coordinate system, distance ${d_{camera}}$ between F and the reference mirror can be calculated by a PnP algorithm [21,22].
A virtual plane is defined to represent the curved screen’s position and posture according to symmetric points displayed on the curved screen. The stereo system in section 2.3 measures its height h and width w. In the stereo system and the deflectometric system, the same virtual plane is chosen to represent the screen’s position and posture.
In the stereo system, ${R_{vp\_stereo}}$ and ${T_{vp\_stereo}}$ denote rotation matrix and translation vector, respectively, between the left camera in Fig. 5 and the virtual plane. In the deflectometry system, rotation matrix and translation vector between the camera in Fig. 3 and the virtual image of the virtual plane in the reference mirror are denoted as ${R_{vp\_virtual}}$ and ${T_{vp\_virtual}}$, respectively. Rotation matrix and translation vector between the camera in Fig. 3 and the virtual plane are denoted as ${R_{vp}}$ and ${T_{vp}}$, respectively. Model of screen at two locations obtained in section 2.3 is denoted as Pcs in the left camera coordinate, and Pmirror in the deflectometry system.
As illustrated in Fig. 6, the following equations transfer the model of the screen from the stereo system to the deflectometric system.
As shown in Fig. 7, the spatial positions of the light source A1, A2, B1, B2 in the screen model corresponding to the pixel of the camera in Fig. 3 are searched by the normal absolute phase reflected by the reference mirror and the deformed absolute phase by SuT. Firstly, the area where the light source located is found by the fringe order constraints. Then, the coordinate of the light source in pixel level is found by monotonicity of the fringe’s absolute phase. Finally, the coordinate of the light source in subpixel level is searched by the bilinear interpolation of adjacent positions’ spatial coordinates and phase values.
Based on the four coordinates corresponding to each pixel of the screen in Fig. 3, height of SuT can be calculated by Eqs. (3)–(6).
3. Experiments and results
In this section, some experiments have been carried out to test the proposed technique by using a curved screen. Firstly, simulated experiments were performed to verify the measurement accuracy. Then, hardware of the deflectometric system has been setup by using a curved screen. Some comparative experiments were performed by using a plane screen and a curved screen. Finally, a plane mirror, a specular step, and a large-curvature specular surface have been measured and reconstructed to test the proposed technique.
3.1 Simulated experiment
3.1.1 Depth of virtual image in the deflectometric system with a curved screen
In order to evaluate the proposed technique by using the curved screen quantitatively, a simulated experiment has been performed. With optical axis being perpendicular to the reference plane, the distance between the camera’s optical center F and the reference plane is d1=600 mm. The axis of a half cylinder, with radio r=300 mm, is perpendicular to zox plane and goes through point O. Point I is the middle of the screen, and the distance between I and the reference plane is IO=500 mm. The point E in the screen reflected by D’ is captured by the camera and its virtual image point E’ is symmetric to E about the tangent line of D’. Radio R of the ideal screen changes from 100 mm to 1500 mm, and R=109 mm represents the plane screen. With the different radios of the curved screen, the virtual image crossed the zox plane is computed. With R=367.521 mm, I’ and E’ are in the same depth, as E’ corresponds to E at the edge of the screen. The depth of virtual image d2 of different screen and its simulation diagram are illustrated in Fig. 8. For any points in screen with the same x, the projection of light to zox plane is also the same. The simulated experiments demonstrate that the curved screen with a proper radio can effectively decrease depth range of virtual image of the fringe patterns.
Fig. 8. Simulation with different screens. (a) simulation diagram, (b) depth of virtual image d2 of different screens.
3.1.2 Measurement of specular surface in the deflectometric system with a curved screen
For testing the proposed deflectometric technique by using an ideal curved screen with a radius of 1000 mm, some simulation experiments have been carried out. A camera was employed to capture the reflected fringe patterns. The reference mirror after translating 20 mm and a specular cylinder with a radio of 60 mm was measured by the simulated system. The distance between screen and the reference mirror is 150 mm. Rotation matrix and translating vector between camera and the reference mirror is $\left( \begin{array}{lcl} 0.9848 &0 &0.1736 \\ 0.0594 &0.9397 &- 0.3368 \\ - 0.1632 &0.3420 &0.9254 \end{array} \right)$ and $\left( \begin{array}{ccc} 0 &0 &378.2081 \end{array} \right)$, respectively.
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=1, …, 15, randn is a matrix whose element obeys standard normal distribution.At different noise levels, the difference of RMS (Root Mean Square) between the measured height by Eqs. (1)–(6) and the real height is illustrated in Table 1 and Table 2.
Table 1. Simulation results of reference mirror after translating
Table 2. Simulation results of cylinder measurement
The simulated experimental results indicate that the proposed technique is of high noise resistance and high reconstruction accuracy.
3.2 Deflectometric system with a curved screen
3.2.1 Hardware of the deflectometric system
The deflectometric system consists of a curved screen, a CCD camera, and a translating stage, as depicted in Fig. 9(b). The camera (XIMEADE XIQ, MQ042CG-CM, Germany) has a resolution of 2048 × 2048 pixels, and uses 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 Qualitative comparisons
In order to explain the advantages of using the curved screen in measuring large-curvature specular surface, deflectometric systems with a plane and a curved screen were setup, as demonstrated in Fig. 9. A cylinder specular surface was tested. The area of displaying fringe patterns for the plane and curved screen was chosen as the same resolution of 1440*900. The pixel size of the plane screen and the curved screen is 0.283 mm and 0.272 mm, respectively, which means the size of the plane screen is larger than that of the curved screen. In the comparative experiments, the plane screen needs to be replaced with the curved screen. Both of two screens were fixed on a translating stage with accuracy of 1 micrometer to be accurately located in a certain position. In order to compare two screens, the specular surface was placed at a position where the camera can capture the largest area of clear patterns.
The camera captured the virtual image of the reflected fringe patterns and the corresponding absolute phase was calculated, as shown in Fig. 10.
Fig. 10. The fringe patterns and absolute phase maps. (a) and (b) with a plane screen, (c) and (d) with a curved screen.
The edge of the virtual image of the fringe patterns displayed by the plane screen is blurred, so the corresponding absolute phase map is noisy on both sides due to out of the depth of field of the imaging lens. However, the virtual image of the fringe patterns displayed by the curved screen is clear because of the small range along depth, as illustrated in Fig. 2. The specular surface was reconstructed by the proposed method with the plane screen and the curved screen and was transferred to same coordinate, as illustrated in Fig. 11. It clearly shows although the physical size of the plane screen is larger than that of the curved screen, the shape data on both sides are missed when using the plane screen, while most parts of the specular surface can be reconstructed when using the curved screen. The qualitative comparison clearly shows that a curved screen can effectively expand the range of gradient and height field of SuT. When a large-curvature specular surface is measured, the small-size curved screen can satisfy with the measuring field view, instead of a large-size plane screen.
Fig. 11. The reconstruction of the specular surface in deflectometric system. (a) with a plane screen, (b) with a curved screen.
3.2.3 System calibration
The radius of the curved screen is 1000 mm in the manual. However, the actual radius is gradually changing, instead of a constant value. In order to calibrate the deflectometric system, the curved screen needs to be reconstructed by using the proposed method in section 2.3. The curved screen displayed sinusoidal fringe patterns with N=10. Two cameras captured the patterns to model the curved screen, as shown in Fig. 12.
Absolute phase maps were demodulated from the captured fringe patterns as shown in Fig. 13. The reconstruction results were fitted by a polynomial surface, as shown in Fig. 14, where the color bar represented depth. Thus, this method was employed to reconstruct the screen at two positions, and the phase value was merged with the spatial point cloud. The result is shown in Fig. 15, where the color bar represents phase.
Fig. 13. Fringe patterns and phase maps. (a)-(d) fringe patterns captured by the left camera and phase map after rectification, (e)-(h) by the right camera.
After reconstructing the model of the curved screen, the deflectometric system was calibrated by using a reference mirror. There are evenly distributed rings, with a radio of 1.5 mm and separation of 6 mm, on the reference mirror. Because aperture value and depth of focus of camera’s lens affect depth of field [23,24], the parameters of the camera lens were adjusted to have depth of focus around 0.3 m and a smaller aperture value. Therefore, markers on the reference mirror and virtual image of the curved screen can be clearly captured. Camera in Fig. 3 recorded signs on the surface of the reference mirror as screen displayed pure white, then recorded virtual image of displayed checker pattern reflected by the reference mirror, as shown in Fig. 16, where + denoted extracted feature points, ○ reprojected feature points. Based on the PnP algorithm, ${R_{vp\_virtual}}$, ${T_{vp\_virtual}}$, ${R_{mirror}}$ and ${T_{mirror}}$ were determined.
Fig. 16. Deflectometric system calibration. (a) Calibration of reference plane, (b) calibration of virtual plane’s virtual image in reference mirror.
Based on Eqs. (9)–(11) and above calibration results, the screen model was transferred to deflectometric system, as shown in Fig. 17, where color bar denoted height of screen to the reference mirror. The reprojection error is [0.1559 0.1044] pixel which computed by virtual plane’s featured points reprojected to the camera by the calibrated parameters in Fig. 17. The process of reprojection refers to literature [25]. The error mainly comes from reconstruction of the curved screen, extraction of markers in reference mirror and virtual plane, and distance of markers on virtual plane which need to be measured by stereo system. Meanwhile, electronic noise from screen and camera will also impact the reprojection error. Eventually, these noise will propagate to reconstruction of the specular surface.
3.2.4 Experiments on specular surface
In order to evaluate the proposed deflectometric system quantitatively, the system measured the plane reference mirror after rotated a random angle. The coordinates of a plane mirror were obtained, as shown in Fig. 18 with the width around 200 mm, the length around 50 mm. As point cloud fitted to a plane, RMS of the distance to the fitting plane is 0.0648 mm.
To verify furtherly, a specular step with high accuracy was measured, as shown in Fig. 19. The actual distance between steps was measured by a three-coordinate measuring machine. Then, deflectometric measured coordinates of every step was fitted to a plane. Finally, the distance between the points of the adjacent step and the fitting plane was treated as measured distance. Table 3 shows the comparison between the actual distance and the measured distance.
Fig. 19. Specular step and deflectometric measurement result. (a) Specular step, (b) deflectometric measurement result.
Table 3. Actual distance, measurement distance of step and error
3.2.5 Quantitative comparisons
A large-curvature specular surface was measured by the proposed technique with a plane and a curved screen illustrated in Fig. 9. The reconstruction of the specular surface through plane and curved screen system were fitted into polynomial surface, as illustrated in Fig. 20.
Fig. 20. The reconstruction of the specular surface in deflectometric system. (a) with a plane screen, (b) with a curved screen.
RMS of the difference between the fitting surface and the reconstruction of specular surface by the plane and the curved screen were 0.0568 mm and 0.0490 mm, respectively. Meanwhile, the range of measurement was expanded by the deflectometric system with the curved screen.
The experiment results show that the proposed deflectometric system can reconstruct specular surface with high accuracy and reliability.
4. Conclusion
In this paper, a novel deflectometric technique has been proposed by using a curved screen to measure large-curvature specular surface. The curved screen effectively expands the range of the gradient and height field of the measured large-curvature specular surface. Firstly, the principle and calibration of the curved screen deflectometric system are given. Stereo cameras are placed in the front of the curved screen to reconstruct the screen. The model of the curved screen is transferred to deflectometric system. Then, a specular cylinder is reconstructed by using a plane screen and a curved screen, and the range of measurement is efficiently expanded by using the curved screen. Finally, a plane mirror, a specular step and a large-curvature specular surface are reconstructed with high accuracy by the proposed deflectometric technique. The experiment results verify the high accuracy and large measured range for measuring large-curvature specular surface by using the proposed deflectometric technique. In the future, further experiments will be carried out to measure more components with large-curvature specular surface and more accurate calibration methods will be studied.
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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