## Abstract

In phase measuring deflectometry (PMD), a camera observes a sinusoidal fringe pattern via the surface of a specular object under test. Any slope variations of the surface lead to distortions of the observed pattern. Without height-angle ambiguity, carrier removal process is adopted to evaluate the variation of surface slope from phase distribution when a quasi-plane is measured. However, in the usual measurement system, the carrier phase will be nonlinear due to the restrictions of system geometries. In this paper, based on the analytical carrier phase description in PMD, a carrier removal method is proposed to remove the nonlinear carrier phase. Both the theoretical analysis and the experiment results are presented. By comparison with reference-subtraction method and series-expansion method, this proposed method can achieve carrier removal process with only the measurement of one single object, as well as high accuracy and time-saving.

© 2013 Optical Society of America

## 1. Introduction

Phase measuring deflectometry (PMD) can be used in measuring the specular surface [1–3], such as the car body, mobile shell, glass and some precision devices surfaces. In PMD, phase distributions in two perpendicular directions are extracted in *x* and *y* direction fringe patterns. The deformed fringe pattern in x direction can be expressed as

*A*and

*B*are the background and modulation intensities, ${f}_{0}\left(x,y\right)$ is the frequency of the carrier fringe and $\phi \left(x,y\right)$ is the shape-related phase angle. Phase extraction algorithms, such as phase shifting method [4], can be used to extract wrapped phase data from several fringe patterns. A phase unwrapping process [5] could subsequently retrieve a continuous phase distribution containing both shape-related phase $\phi \left(x,y\right)$and carrier fringe-related phase components$2\pi {f}_{0}\left(x,y\right)x$. Therefore, the carrier must be removed from the overall phase distribution for evaluation of the phase of interest$\phi \left(x,y\right)$.

Carrier removal is not an essential step in the common case of PMD because it is usually incorporated in the system calibration. The height-angle ambiguity problem is one of the difficulties in PMD. To solve this problem, analytical fitting methods and geometric constraints enhancing methods are proposed. However, accuracy of calibration parameters limited the improvement of the accuracy, and the request of mechanical shifting and geometrical restrictions of the time-consuming calibration reduced the flexibility of these techniques.

On the other hand, the ambiguity can be neglected if a quasi-plane with very small depth variation is measured. In other words, the phase is only modulated by the local slope without disturbance of the depths. Conversion of the measured phase distribution to the object slope distribution is required and topography of the specular object is reconstructed by numerical integration. Thus, carrier removal methods can serve as an easier solution to the phase-to-slope translation when comparing with the geometric constraints enhancing methods with time-consuming calibration.

In optical surface measurement technology, system arrangement is usually adjusted carefully to help remove nonlinear carrier components and guarantee measurement accuracy, but nonlinear carrier components can hardly be removed completely. In fringe projection profilometry, several carrier removal methods [6–9] have been studied, namely spectrum-shift, average-slope, plane-fitting, reference-subtraction, Zernike polynomials method and series-expansion method. The latter three methods can serve as alternative methods in removing nonlinear carrier phase in PMD.

Our latest work on the analytical description of carrier component and reference subtraction technique for carrier-removal in the PMD is presented in [10]. The reference paper indicated that reference-subtraction method needs to measure the specular surface and the reference plane separately, so the measurement uncertainty is magnified. What’s more, reference-subtraction method requires the object and the reference mirror to be placed in the same position in the two separate measurements, which will introduce placing error to the measurement when this condition cannot be satisfied.

In this paper, a carrier removal method based on the analytical carrier phase description is proposed in PMD for the first time. In section 2, the principle of the method is discussed. Computer simulations and experimental work of nonlinear carrier removal process are presented in section 3, and comparison of reference-subtraction method, series-expansion method and the proposed method is investigated. Conclusions are drawn in section 4.

## 2. Principle of the proposed method

Figure 1 shows the arrangement of the PMD. Computer generated phase-shifting fringe patterns in *x* and *y* directions with an equal spacing period P are displayed on a liquid crystal display (LCD) screen. The CCD camera records the distorted fringe patterns reflected by the specular surface.

Equations (2) and (3) show the relation between phase and slope in *x* and *y* directions, in which ${L}_{y}$ presents the distance from LCD screen and reference plane, ${\phi}_{rx}$and${\phi}_{ry}$are carrier phase in the two directions, ${g}_{x}$and ${g}_{y}$ are two components of slope of the surface. So, only the carrier phase ${\phi}_{rx}$and ${\phi}_{ry}$are removed precisely can we get the accurate phase-to-slope relationship.

Figure 2 shows the geometrical arrangement of PMD. In the figure, *xyz* and *XYZ* are defined as the world coordinate and local coordinate of CCD plane, (*X _{f}*,

*Y*) is the optical center of the lens of the CCD camera. The optical axis of the CCD camera crosses the imaging center (

_{f}*X*,

_{0}*Y*) perpendicularly. The LCD plane is vertical to

_{0}*xz*-plane, making an angle

*θ*with

*xy*-plane.

*P*(

*x*,

_{0}*z*) is the original point on LCD, where the phase is set to zero. The light reflected by any point

_{0}*C*on the specular surface, received by CCD pass through the optical center, can be traced back to its source location on

*B*or

_{x}*B*to calculate the phase distribution.

_{y}The relationship between carrier phase and slope distribution of the specular surface in both horizontal and vertical directions can be obtained by analyzing the geometrical arrangement of PMD in Fig. 2. Equations (4) and (5) [10] are the analytical expressions of carrier phase in PMD in horizontal and vertical directions, which demonstrate the fact that the carrier phase distribution is nonlinear. The x direction carrier phase is described as a function of x direction spatial variable, whereas the y direction phase distortion is modulated by two spatial variables.

It should be noticed that Eqs. (4) and (5) are only valid when the camera imaging plane is perpendicular to *xz*-plane and *Y*-axis is parallel with *y*-axis. Assume the CCD is rotated around *Z*-axis and the optical center of imaging lens (*Xf*, *Yf*) in *YZ*-plane by angles φ and ρ, respectively. The *x*-direction fringe pattern is not imaged as pure vertical fringe pattern but tilted according to the rotation. The image mapping transformation is modified according to the rotation and shifting of the camera coordinates *XYZ* in the world coordinates *xyz*.

After combining the system parameters, Eqs. (4) and (5) become Eqs. (6) and (7).

2-order Taylor expansion is used to unfold the fractional part of Eqs. (6) and (7), and the results are shown in Eqs. (8) and (9). It should be noted that subsequent improvement by raising the order would be smaller than phase measurement uncertainty.

Equations (8) and (9) are the fitting expressions of carrier phase removal in x and y directions of the proposed method. Least-square method is used to solve fitting coefficients. Data points used for the estimation of the carrier function should be in the vicinity of the reference plane. The proposed method demands that the sampling regions and the sampling data points from the reference plane are enough to fit the carrier phase, which requires the sampling regions cannot concentrate upon a small part of the reference plane. Other requirement of positioning in the reference plane is not necessary. The flow of the proposed method is shown in Fig. 3.

Since the proposed method is used to remove the carrier phase from the overall phase distribution, the slope and height variation of the object should meet the requirement that the wrapped phase could be unwrapped successfully. It should be noted that maximum slope and height variation really have some influence over PMD after removing carrier frequency. The slope-phase relation equations (Eqs. (2) and (3)) are obtained under the assumptions that the test object is quasi-plane and the test object has small slope variation. If the conditions cannot be met, big errors will be introduced into PMD.

## 3. Computer simulation and experimental results

Computer simulations were carried out to test the performance of the proposed carrier removal method. The carrier phase distributions in x and y directions were Eqs. (10) and (11), which obey the forms in Eqs. (4) and (5). The slope related phase was parabolic, and the unwrapped phase distributions which contain carrier phase and slope related phase in x and y directions are shown in Figs. 4(a) and 4(b). The proposed method was used to remove the carrier phase, and Figs. 4(c) and 4(d) show the error distributions of proposed method.

The RMSE (Root-Mean-Square-Error) of the proposed method in x and y directions are 0.026 and 0.009. It should be noted that error distributions of 2-order series-expansion method are the same as the proposed method, which means the coefficients of rest 12 terms in 2-order series-expansion method are near zero. Figure 4(e) shows the reconstruction height error. The error of the reconstructed height data is related to the 3-order and higher order components of nonlinear carrier phase.

Measurement setup of the experiment is shown in Fig. 5. The hardware system is composed of a CCD camera (Manta G-125B/C) and a LCD screen (PHILPS 170S9). The camera uses a 50mm focal length lens (Computar M5018-MP2).

To verify the performance of the proposed method in actual experimental system, a plane mirror was measured. According to Fig. 2, the plane mirror was placed 479.0mm away from the LCD screen and 500.0mm away from the CCD camera. The angle between LCD plane and xy-plane θ was 17.2°. The fringe pitch on the LCD screen was 10.0mm and the focal length of camera lens was 50mm. Figure 6 shows the procedure of the carrier removal in x direction.

The error distribution indicates that the carrier phase contains a y spatial variable related component, which is inconsistent with Eq. (4). That’s because Eqs. (4) and (5) are only valid when the camera imaging plane is perpendicular to *xz*-plane and *Y*-axis parallel *y*-axis.

It can be seen from the experimental results that the influence of rotation matrix and shifting matrix on carrier phase removal accuracy is small. What’s more, improvement of the form which is shown in Eqs. (12) and (13) may eliminate the influence. Figures 6(d) and 6(e) show the error distributions in x and y directions after improving the form. Figure 6(f) is a reconstruction of the plane mirror, and RMSE of the reconstruction is 0.38μm.

A mirror with scratches on the surface was measured, and Fig. 7(a) shows the mirror and the scratches. The proposed new method was used to remove carrier phase to reconstruct the surface shape of the denoted region of Fig. 7(a), and the result is shown in Fig. 7(b). Comparison of a column(x = 4.0mm) of reconstructing results by using proposed method and 2-order series-expansion method separately is shown in Fig. 7(c), which demonstrates the accuracy of the two methods in removing the nonlinear carrier phase in PMD is similar. Reference subtraction method was used to remove carrier phase to reconstruct the surface shape of the denoted region of Fig. 7(a), and Fig. 7(d) shows the reconstructed shape. There exists a large error in the reconstruction shape. The reason is that the placing error affects the reconstructed shape. In the experiment, it is hard to place the reference plane and the test object in the same position in the two separate measurements.

Table 1 compares the accuracy (represented by Root-Mean-Squared-Error of error distribution) and the consuming time of 2-order series-expansion method, the proposed method and reference subtraction method in removing carrier phase when the fringe patterns is in x direction. The time was calculated with computer of Inter Pentium CPU G620 @ 2.60GHz and 2986MB RAM, and the number of the sampled dates was 400.

In comparison with reference subtraction method, our method needs only the measurement of a single object and has a high accuracy, and compared with series-expansion method, the proposed method has a simpler fitting form and relatively short processing time.

## 4. Conclusion

In an actual imaging system, non-telecentric imaging would lead to a nonlinear carrier, which must be removed accurately to get the slope-related phase. In this paper, a new carrier removal method based on the analytical carrier phase description in PMD is proposed. Computer simulation and experimental work indicate its good performance in removing the carrier phase in PMD. By comparison with reference-subtraction method and series-expansion method, this proposed method can achieve carrier removal process with only the measurement of one single object, as well as high accuracy and time-saving.. The new method is expected to be used in real time or on line measurement of specular surface.

## Acknowledgment

The authors wish to acknowledge the support by the Fundamental Research Funds for the Central Universities (No.ZYGX2011J053), National Nature Science Foundation of China (No. 60925019, 61090393) and Science and Technology Innovation Team of Sichuan Province (No. 2011JTD0001).

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