## Abstract

Phase measuring deflectometry (PMD) is an effective technique for three-dimensional measurement of specular surfaces. However, the ambiguity of monoscopic PMD and the time-consuming searching process of stereoscopic PMD are challenges for specular surface reconstruction. To solve it, we propose an iterative reconstruction algorithm for the stereoscopic phase measuring deflectometry system free of the time-consuming searching process for each pixel. An arbitrary seed point on the specular surface is accurately obtained via a coarse-to-fine optimization means without any other expensive and complicate auxiliaries. Then, a plane with the height of seed point is set as the initial surface form for the iteration, in which the pinhole model is used to find the linear relation to update the surface form. The converging height is the output as the final result. Simulations and experiments verify the feasibility and efficiency of our proposed method based on the stereoscopic phase measuring deflectometry system. The accuracy and robustness are comprehensively evaluated as well.

© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

Three-dimensional (3D) reconstruction is of great significance in industrial applications. Fringe projection (structured light) is widely used in the reconstruction of diffuse surfaces [1–3]. But for the surface with specular properties, such as metal polished surface, mirror surface and transparent glass, structured light projection method cannot work well. Due to the reflective property of the surface of high-gloss objects, the measurement of their 3D surface morphology is always a challenge. The existing optical non-contact methods for measuring mirror objects are mainly divided into interferometry and deflectometry [4,5]. The interferometry obtains the distance information with the phenomenon of interference [6,7]. It has high measurement accuracy and resolution, but it is unable to measure objects with large curvature surface. Even for the free-form specular surface, it is difficult to reconstruct. Moreover, the complicated operation and the exorbitant price limit its application. The phase measuring deflectometry can break the limit and restore the 3D morphology of the specular object via geometric relationship between the deflection of the light and the surface under test (SUT). The PMD has become a research hotspot due to its advantages of simple principle, low cost, high dynamic range and full-field measurement in measuring mirror objects [8–10]. For a typical PMD system, the fringe patterns generated by software are displayed on a screen, namely a liquid crystal display (LCD), then the camera (CCD) captures the deformed reflected patterns, which is decoded to extract phase information for the corresponding pixels between the LCD and CCD. The PMD is a specular 3D measurement technique based on slope calculation, then slope integration [11–13] is adopted to recover the 3D shape data of the SUT. The above principle is applicable for both monoscopic PMD (mono-PMD) and stereoscopic PMD (stereo-PMD), but different challenges are encountered in each case.

For a mono-PMD system, the height-normal ambiguity is ubiquitous due to the fact that the phase information is modulated by both the height and the slope of the SUT. To solve the ambiguity, different techniques have been investigated. One classical solution is to either move the screen [14,15] or apply a plate beam splitter to realize the parallel design of two LCD screens, which was named as direct PMD (DPMD) [16,17]. The DPMD determines the ray from the LCD to the SUT and obtains the surface directly by triangulation. However, such physical displacement deteriorates the measurement precision if we consider component displacement errors, so it has strict requirements on operating accuracy. Similar to DPMD system, Li et al. proposed a bi-plane PMD scheme, where the bi-plane is obtained by directly covering a transparent panel above the LCD screen [18]. Another solution is assuming a prior height distribution, rough shape and position of the measured surface is required to be estimated. Su et al. developed a software-configurable optical test system to measure telescope mirrors based on the advanced model [19,20]. Huang et al. further improved mono-PMD by applying a well-established mathematical model [21]. Moreover, Graves et al. presented a novel model-free iterative data-processing approach for deflectometry tests of unknown surfaces with a seed point located by a coordinate measurement machine (CMM) [22,23]. Similarity, Li et al. recovered the specular surface with the help of a confocal white-light distance sensor or the laser sources [24]. The auxiliary equipments are expensive and need to be carefully calibrated with the PMD system. In addition, some calibration optimization algorithms [25–27] and error compensation [28–30] means can also relieve ambiguity and improve accuracy.

For the stereo-PMD system, matching corresponding points is the major challenge. Different from traditional binocular vision, the feature consistency cannot be acquired from the images. The correspondence is searched by the consistency of two normal vectors that calculated by the Snell's law. One method is searching correspondences along the epipolar line on the image [31] and another common solution is searching the surface point in 3D space [8,32–34]. Recently, Zhang et al. proposed a novel method to find the correspondence by minimizing the phase difference on the LCD instead of the angular difference of two normal vectors [35]. However, it spends a lot of time to find the correspondence for each pixel.

In consideration of the need of auxiliary to position the seed point in the mono-PMD as well as the time-consuming of the stereo-PMD, we propose a novel iterative reconstruction algorithm for stereo-PMD without the expensive and complicated auxiliary. An arbitrary seed point on the SUT is accurately obtained via a coarse-to-fine optimization means. Then a flat surface with the height of seed point is defined as the initial input for the iteration, in which the coordinate of the seed point is used to define the piston of the SUT because the integration algorithms reconstruct the relative surface rather than the absolute height. The pinhole model is used to established a liner relation to update the SUT iteratively, avoiding the complex surface segmentation in model-free framework [22,23]. Finally, the converging surface is output as the result of the SUT.

The outline of this paper is as follows. Section 2 introduces the principle of the proposed method in details. The simulation is carried out in MATLAB to verify the feasibility of the method in section 3. Section 4 shows the practicability of the method through actual measurements. The conclusion is given by the last section.

## 2. Principle

As shown in Fig. 1, the general stereo-PMD system consists of two cameras symbolled as *O _{l}* and

*O*, an LCD screen and the SUT. Structured light patterns (SLPs) displayed on the screen are reflected by the specular surface, and captured by both cameras. In other words, the cameras capture the virtual image of the real screen. The correspondence between each camera pixels and the screen pixels are matched by decoding the SLPs. As for the correspondence of the pixels

_{r}*p*and

*q*in two cameras, the same surface point

*M*shows different features in two cameras, which is the distinct difference from binocular vision. Therefore, we turn to 3D space instead of the 2D image. To find the correspondence, the angular difference of two normal vectors

**and**

*n*_{l}**is treated as the cost function for matching.**

*n*_{r}#### 2.1 Position of the seed point

Different from the previous works [22–24] which positions the seed point with auxiliary equipment in mono-PMD system, we use the normal consistence of the stereo-PMD to locate the seed point. The seed point is used for initialization and determining the piston of the SUT in iteration, therefore, the precision should be guaranteed. To this end, a coarse-to-fine optimization algorithm is investigated to avoid the local converge. For the first step, the classical stereo-PMD technique proposed by Knauer et al. [8] is applied to get the coarse position. As shown in Fig. 1. Space points are searched along the probe ray by left camera. Suppose *M* to be the seed point on the SUT, viewed at pixel *p* on left camera and pixel *q* on right camera. The corresponding screen point *p’* and *q’* on the LCD are located by the phase information. Assuming the stereo-PMD system has been calibrated, the incident rays ** l**,

**and reflected rays**

*r***,**

*m***of each camera can be acquired. Two normal vectors at the point**

*s**M*are calculated by the Snell's law respectively:

The angle *α* of ** n_{l}** and

**is:**

*n*_{r}Theoretically, the angle *α* should be zero when the searched point is actually on the SUT in condition that the system calibration and phase information is precise enough. Practically, the normal vectors calculated from these two cameras cannot exactly have the same direction due to various errors. Suppose the searching domain is [*M*_{1}, *M*_{2}] with evenly nodes of this region, and a series of *α* is obtained by Eq. (2). The corresponding surface point of the minimum *α* is marked as *M _{coarse}*. Then to improve the accuracy,

*M*is set as the initial input for a nonlinear optimization. The objective function is further minimizing the

_{coarse}*α*and converging to the fine-toned point

*M*in the neighborhood of

_{fine}*M*.

_{coarse}*M*is regarded as the seed point for the following iterative reconstruction. The procedure is shown as Fig. 2.

_{fine}#### 2.2 Iterative reconstruction algorithm

To reconstruct the specular surface, the iterative sense in mono-PMD is referred. Consider the left camera and the LCD as a mono-PMD system in Fig. 3. A ray from the pixel point *P *= (*x _{p}, y_{p}, z_{p}*) goes through the camera optical center

*O*(

_{l} =*x*), reflected by the SUT at

_{c}, y_{c}, z_{c}*T*= (

*x*), and hits the screen at point

_{t}, y_{t}, z_{t}*S*= (

*x*). The coordinates are all unified to the world coordinate system {w} via the system calibration. Then the x-slope

_{s}, y_{s}, z_{s}*s*and y-slope

_{x}*s*are calculated by Eq. (3) [25]:

_{y}*d*and

_{t2c}*d*are the distance from the measured point

_{t2s}*T*to the camera center

*O*and corresponding screen pixel

_{l}*S*, respectively. The surface height to be measured is marked as

*h*. To start with, a plane with the height of the seed point

*M*is regarded as the initial surface form (

_{fine}*h*°) of SUT. The probe rays

**= (**

*l**x*) can be calculated by the camera intrinsic parameters and transformed to the world coordinate system. The intersection of the rays and the initial plane is marked as

_{l}, y_{l}, z_{l}*T*°

*($x_t^0,y_t^0,z_t^0$), which is calculated by Eq. (4):*

^{ }=*c*and

_{i}*Z*are

_{i}*i*-th term of the coefficients and the Zernike polynomials. The constant term

*c*

_{0}indicates the piston of the surface. To obtain the coefficients in Eq. (5), the analytic derivatives of

*h*should be equal to the measured slope data:

The number of basis function *N* is adjustable to approach the SUT. The Eq. (6) is overdetermined and can be solved in a least-squares sense. Using the solved coefficients *c _{i}*, we can apply the interpolant with Eq. (5) to reconstruct the SUT. The seed point

*M*= (

_{fine }*x*) is used to calculate the

_{m}, y_{m}, z_{m}*c*

_{0}as follows:

The reconstruction height *h* is regarded as the updated SUT for the next loop. Similar to Eq. (4), for the *n*-th iteration, the surface point ${T^n} = (x_t^n,y_t^n,z_t^n)$ is updated as:

Then the slope calculation with Eq. (3) and the integral reconstruction with Zernike polynomials are executed repeatedly. The converging height *h ^{n+}*

^{1}is output as the final result of the SUT as shown in Fig. 3. The iterative flowchart is shown in Fig. 4.

## 3. Simulation

To verify the feasibility of the proposed stereo-PMD iterative algorithm, a numerical simulation is conducted to reconstruct a concave and a convex surface. The stereo-PMD model is established as shown in Fig. 5. The intrinsic parameters of a real calibrated camera are assigned to the two simulated pinhole camera models. The resolution is 2080×1552 pixels with the pixel pitch of 2.5 um. The lens has a focal length of 12 mm. The optical axes of the two cameras are set to be parallel with a distance of 80 mm. The world coordinate system is located at center of the common field of view (FOV) with a distance of 450 mm to the red camera in Fig. 5. The screen is put on the proper position and the relationship with respect to the cameras are assigned.

Once the system parameters of the simulation model are determined, a concave and a convex sphere surface with a radius of curvature (ROC) of 500 mm are generated as the SUT (Fig. 6(a) and Fig. 6(d)). Their peak and valley are positioned at the coordinate axis center, respectively. The range size is a square with sides of 50 mm. To simplify the simulation process, the reverse ray tracing is applied to find the correspondence between the screen points with each camera pixel. The probe rays’ directions are determined by intrinsic parameters with pinhole model, and the corresponding points on the SUT and the screen are obtained by the intersection of the line and plane with ray tracing method. Then the proposed iterative method shown in the flowchart of Fig. 4 is used for reconstruction. The absolute errors between the reconstruction results and the real SUT are shown in Fig. 6(b) and Fig. 6(e), and the root mean square (RMS) is 9e-4 nm and 1e-3 nm, respectively. The fitted ROC is equal to the real ROC of 500 mm.

Then, to better compare with the real situation, the performance with measurement noise on phase is also investigated under the same simulation condition. Normally distributed random noise with standard deviation of 0.1 pixel is added onto the analytical phase map. The reconstructed errors with noise are shown in Fig. 6(c) and Fig. 6(f). The errors of RMS are 0.9 nm and 1.5 nm, respectively. From the simulated results, it is proved that the proposed method is feasible to perform specular surface measurement in the stereo-PMD system.

## 4. Experiment

To verify the performance of the proposed method in practice, a stereo-PMD system is established as shown in Fig. 7. The system consists of a 13.3-inch portable display and two industrial cameras (FLIR BFS-U3-32S4M-C). The display has a resolution of 1920 × 1080 and the pixel pitch is 0.1534 mm. Both the cameras have the resolution of 2048 × 1536 pixels and the pixel size is 3.45 um, the focal length of the lens is 12 mm. Distance from the SUT to the cameras is about 260 mm. To be fair, each measurement is conducted in MATLAB 2016b of the same PC platform (Intel Core i7-4790CPU @3.60GHz 16GB RAM).

The system should be calibrated before the measurement. Three terms are considered for system calibration, such as camera parameter calibration and system geometric calibration as well as the gamma nonlinearity calibration. Previous work [36] of mono-PMD calibration is extended to this stereo-PMD system. Due to the nonlinear response of digital devices such as screens and cameras, a nonlinear response curve of screen and camera is obtained experimentally [30]. Then, image intensity will be compensated for the captured patterns to reduce the phase error.

After the stereo-PMD system is calibrated, two concave and two convex reflective mirrors with different ROC and diameter (*Φ*) are measured with the proposed method. The SLPs are sequentially displayed on the LCD and simultaneously captured by two cameras. Two of horizontal and vertical patterns captured by both cameras are shown in Fig. 8. Firstly, the gamma correction is conducted for the sinusoidal patterns with the gamma calibration. Then captured SLPs are decoded for horizontal and vertical phase maps to match the CCD pixel with the LCD pixel. To facilitate calculation, systematic calibration parameters are used to position these corresponding points and transform to the world coordinate system. An arbitrary pixel in left camera is chosen as the correspondence of the seed point *M _{fine}*, which is positioned by the procedure in section 2.1. At last, the SUT is reconstructed via the iterative algorithm illustrated in section 2.2.

A concave mirror of *Φ = *25 mm with ROC = 304.8 mm is reconstructed in the first row (a-c) of Fig. 9. The 3D model is shown in Fig. 9(a) with the z-axis zoom in and the Fig. 9(b) is the planform of the result. By fitting the point cloud with a sphere, the fitted radius is 303.6 mm. The fitting error map is given in Fig. 9(c), the peak and valley (PV) and the RMS are 380 nm and 69 nm respectively. As we notice that the error map is bended at the edge, where such situation appears in other measurements. We attribute it to the decoding error as the fringe information is not accurate at boundary. Therefore, parts of edges are cut out and the range of result is smaller than the real SUT.

For comparison, above concave mirror is also reconstructed by the classical stereo-PMD method, in which the correspondence of each pixel of two cameras should be established. In this case, the searching process of space points spends a large amount of time. The result is shown in the second row (d-f) of Fig. 9. The fitted ROC is 303.8 mm and the RMS is 89 nm. Though the accuracy is very close, the computing time is sharply decreased with our method. In the experiment, it takes 1779 seconds to complete the measurement while our proposed method takes only 72 seconds. Apparently, our method performs well in both precision and speed.

Then a concave mirror of *Φ = *50 mm with large ROC = 1000.0 mm is measured. The result is shown in the first row (a-c) of Fig. 10. The fitted radius is 1006.2mm, as the ROC is so large that little surface error can lead to large radius error. Even though, it is within the ROC tolerance ${\pm} $ 2% provided by the manufacture. The error map is given in Fig. 10(c). The PV and RMS are 1200 nm and 197 nm, respectively.

Later on, two circular convex mirrors with different ROC of 105.0 mm and 456.0 mm are measured as well. The second and third row of Fig. 10 represent the results and error maps of two convex SUT, respectively. The reconstructed RMS of errors are 275 nm and 456 nm.

For convenience, the nominal ROC and the measured ROC as well as the fitting errors of the four SUT are listed in Table 1.

Apart from the reconstruction accuracy, the robustness of the system is evaluated by repeated measurements as well. Each SUT is measured three times and the average surface is obtained by averaging the three results. As shown in Fig. 11, the first column represents the average surface and the difference between the average shape and individual result is given by the remaining three columns. The PV and RMS of each difference map (piston, tip and tilt removed) illustrate the robustness of the proposed method.

## 5. Conclusion

In this paper, a novel iterative algorithm for stereo-PMD is proposed to measure the specular surface. On the one hand, the ambiguity of the mono-PMD is resolved and free of the other detection auxiliary equipment with high cost. On the other hand, the time-consuming searching process in stereo-PMD system is avoided and reconstruction is accelerated. For general mono-PMD technique, the iterative reconstruction with prior shape model or a seed point depends on the user input. Whereas in our method, the seed point is obtained and input to the iteration loop automatically. In addition, not the plane of x-o-y but the height of the seed point is set as the initial surface form, which decreases the number of the iteration apparently, especially when the SUT is far away from the plane of x-o-y. Meanwhile the accuracy of the proposed algorithm is well illustrated by the Table 1. The fitted ROC of four SUT are closed to the nominal ROC with manufacturing tolerance of ±2%. For the concave of Φ = 25 mm, the reconstructed accuracy is approaching nanoscale. Compare with the classical stereo-PMD, the proposed method stands out in both efficiency and accuracy. Moreover, the robustness of the system is also proved by the repeated measurements. The PMD measurement principle depends on the basic Law of Reflection, and the surface normal is determined by performing an inverse ray tracing from the camera pixel to the LCD pixel. The method can be applied for surfaces like mirrors, metal polished and other material surfaces as long as they satisfy the Law of reflection. But for the transparent objects like glass, lights can travel through the front surface by refraction and reach to the back surface, complex light paths make it hard to reconstruction. Further study will focus on the measurement of the transparent glasses as well as the registration for the stereo-PMD and multi-camera PMD system.

## Funding

National Natural Science Foundation of China (61775172); Key-Area Research and Development Program of Guangdong Province (2019B010149002, 2020B090925001); Natural Science Foundation of Shenzhen City (JSGG20191129114035610).

## 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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