## Abstract

Fringe projection profilometry is generally used to measure the 3D shape of an object. In oblique-angle projection, the grating fringe cycle is broadened on the reference surface. A well-fitted, convenient, and quick cycle correction method is proposed in this study. Based on the proposed method, an accurate four-step phase shift method is developed. Comparative experiments show that the fringe projection profilometry based on the novel phase shift method can eliminate cycle error and significantly improve measurement accuracy. The relative error of the measurement is less than 1.5%. This method can be widely employed for measuring large objects.

© 2011 OSA

## 1. Introduction

3D shape measurement is widely applied in production automation, robot vision, virtual reality, medical image diagnosis, and other fields [1,2]. Fringe projection profilometry offers the advantages of non-contact operation, full-field acquisition, high resolution, and fast data processing. For these reasons, fringe projection profilometry has become one of the most important 3D shape measurement methods [3,4]. However, despite the widespread use of the intersecting axis measurement system, it has several limitations. First, the original standard sinusoidal grating is deformed after oblique-angle projection, and the cycle of the grating fringe on the reference surface is non-uniform and broadened. The fringe cycle is not a fixed value and is always obtained through the average method. Thus, error is always expected, and measurement accuracy is reduced. In measuring large objects, measurement accuracy decreases with increasing distance from the origin. Second, the broadening of the fringe cycle on the reference surface is not considered in the traditional phase shift method. The fringe phase shift on the projection surface is probably the same as that on the reference surface. However, dissimilarity exists because of the oblique-angle projection. Thus, an accurate phase shift method must be developed.

Because the fringe cycle on the reference surface is an important parameter, a number of researchers have conducted in-depth studies on this parameter. Consequently, several solutions have been proposed for its improvement. Gorthi and Rastogi [5] provided a detailed description of the theory, application, and limitations of fringe projection technology. Wang and Du [6] determined the relationship, as well as the modified height expression, between the fringe cycle of the virtual reference surface *x′ *and that of the reference surface. Sansoni *et al.* [7] demonstrated that in intersecting axis systems, the original standard sinusoidal grating is not a fixed value after being projected to the reference surface when an oblique-angle projection is used. If the fringe cycle on the reference surface is taken as a fixed value, height errors occur. An error compensation algorithm is proposed as a corrective measure. Zhang *et al.* [8–10] proposed a novel uneven fringe projection technique to obtain evenly spaced fringes in the measurement volume. A simple, flexible calibration method was also proposed. Chen and Quan [11,12] and Wang and Bi [13] determined the relationship between the fringe cycle of the projection surface and that of the reference surface. The authors also obtained the phase on the reference surface through the least-squares method. Cheng *et al.* [14] analyzed the situation of orthographic projection and oblique-angle reception, and obtained the height expression of the measured object. Maurel *et al.* [15] pointed out that fringe cycle varies from 0.35 to 0.4 cm from left to right. The expression of fringe cycle on the reference surface is obtained when an oblique-angle projection is employed. Rajoub *et al.* [16] obtained the relationship between the phase and the height of a measured object based on geometric analysis. Salas *et al.* [17] obtained the mathematical relationship of three coordinates, namely, projection, object, and camera coordinates. A mapping relationship between the 3D shape and the phase is obtained independently on an equivalent wavelength. This process eliminates the measurement error caused by variations in wavelength. Liu *et al.* [18] and Fujigaki *et al.* [19] proposed a look-up-table approach to calibrate the system where the height should be precisely controlled during the calibration procedure to provide every pixel the relationship between its phase values and dimensions. Srinivasan *et al.* [20] proposed a phase-mapping approach to remove the nonlinear carrier. The least-squares method was used by Huang *et al.* and Vo *et al.* [21,22].

The above mentioned methods require the determination of a number of variable parameters or angles. These methods can correct the errors and improve measurement accuracy. However, these processes are either complex or cannot completely eliminate the error of a grating fringe cycle. Thus, a new correction method is necessary to fix the errors caused by oblique-angle projection. The method presented in this paper can correct the fringe cycle on the reference surface and transform it into a constant. Based on this method, an accurate four-step phase shift method is proposed.

## 2. Principle

#### 2.1 Theoretical model for cycle correction of grating fringe

In Fig. 1
, *O*′ is the optical center of the projector. *X* is the reference surface, *X*′*is the virtual reference surface, and X′′is the surface of the projector. L is the distance between the camera and the reference plane, and d is the distance between the camera and the projector. The projector and the camera have the same distance in relation to the reference surface. The grating fringe is on an oblique angle projected to the surface X. The fringe cycle is broadeningon the reference surface. X′ is perpendicular to the central axis of the projector. OA = x and OA′= x′ are assumed. Consequently, $\frac{x\text{'}}{x}=\frac{\mathrm{sin}({90}^{\circ}-(\alpha +\theta ))}{\mathrm{sin}({90}^{\circ}+\theta )}=\frac{\mathrm{cos}\alpha \mathrm{cos}\theta -\mathrm{sin}\alpha \mathrm{sin}\theta}{\mathrm{cos}\theta}$. Subsequently,*

*$$x\text{'}=x\mathrm{cos}\alpha -x\mathrm{sin}\alpha \mathrm{tan}\theta =\frac{xL{({L}^{2}+{d}^{2})}^{1/2}}{{L}^{2}+{d}^{2}+d\cdot x},$$*

*where$\alpha =\mathrm{arctan}\frac{d}{L}$, and $\alpha $is the projection angle of the projector. $op={({L}^{2}+{d}^{2})}^{1/2}$. The following formula is further derived:The fringe cycle is supposedly uniform on the surface$$\phi (x\text{'}\text{'})=\frac{2\pi f\cdot o{p}^{2}M\cdot x\text{'}\text{'}}{L\cdot op-M\cdot x\text{'}\text{'}\cdot d}.$$Clearly, $\varphi (x\text{'}\text{'})$is a hyperbolic function.**X*; hence, the fringe cycle is certainly non-uniform on the surface*X*′ and on the projector surface. Given that the fringe phase is $\phi (x)=2\pi fx$ on the surface*X*, where$f$ is the grating frequency on the surface*X*, the phases of points*A*and*A*′ are shown to be the same. Subsequently, the phase on the surface*X*′ is described as*M*is the magnification of the projector, and$x\text{'}=Mx\text{'}\text{'}$, where $x\text{'}\text{'}$ is the coordinate position on the projector surface. The fringe phase distribution on the projector surface is described as#### 2.2 Simulation experiment of cycle correction

To obtain a grating fringe with a non-uniform cycle on the projector surface, the phase relationship in Eq. (4) is used to acquire the grating generation software. *M* is set to 2, *L* to 119 cm, *d* to 33 cm, and $f$to 2. The phase curve is shown in Fig. 2(a)
. The sinusoidal is drawn according to the phase curve, as shown in Fig. 2(b). Subsequently, the projection grating is drawn according to the sinusoidal, as shown in Fig. 2(c). As a result, the projection grating with a non-uniform cycle is produced. The cycle of the grating decreases from left to right. A grating with a uniform cycle is obtained after projecting the grating to the reference surface, as shown in Fig. 2(d). Figure 2(e) is the spectrum of Fig. 2(c). Figure 2(f) is the spectrum of Fig. 2(d). Figure 2(e) shows numerous spectrums, whereas Fig. 2(f) presents only one that shows that the fringe cycle in Fig. 2(d) is uniform.

#### 2.3 A novel phase shift method

For the ordinary four-step phase shift method, the electronic grating on the projection surface is generated by a computer. The grating fringe cycle is uniform, and phase shift is easy to achieve. For example, a 90° phase shift is achieved when the grating fringe on the projection surface moves a quarter of a cycle. In the present paper, the grating fringe cycle on the projection surface is non-uniform, and thus directly achieving a four-step phase shift is difficult, especially if the prerequisite of the grating fringe cycle on the reference surface is corrected as a constant. If the grating fringe on the reference surface is moved by 0, *T*/4, *T*/2, and 3*T*/4, where *T* is the grating cycle, the corresponding phase shifts are 0°, 90°, 180°, and 270°, respectively. Thus, if the grating fringe on the reference surface is moved by *T*/4, the coordinate on the projection surface must be obtained. The coordinate on the projection surface is obtained using Eq. (2): $x-T/4=[o{p}^{2}x\text{'}/(L\cdot op-x\text{'}\cdot d)]$.

If $\varphi (x)=\varphi ({x}^{\prime})$, then$\varphi (x\text{'})=\frac{2\pi f\cdot o{p}^{2}\cdot x\text{'}}{L\cdot op-x\text{'}\cdot d}+2\pi f\frac{T}{4}$ and $\varphi (x\text{'}\text{'})=\frac{2\pi f\cdot o{p}^{2}M\cdot x\text{'}\text{'}}{L\cdot op-M\cdot x\text{'}\text{'}\cdot d}+2\pi f\frac{T}{4}$.

Generally, a four-step phase shift is achieved on the reference surface, and the phase distribution on the projection surface is obtained using the following formula:

## 3. Experiments

The experiments are conducted based on the principle of Fig. 1. After the verticality and parallel calibration of the system [23], standard blocks are used to calibrate the parameters *L* and *d*. *L* is 159.004 ± 0.022 cm and *d* is 74.002 ± 0.017 cm. *M* is obtained through a simple experiment, and *M* is 3.442 ± 0.008. $f$can be set to a reasonable value depending on the requirement. *L*, *d*, *M,* and $f$can be set from the input interface of the grating generating software. Subsequently, the desired grating can be achieved. For example, when *N* = 0, $f$is set to 1/2.860 cm^{−1}. Based on Eq. (5), $\varphi (x\text{'}\text{'})=$204.009 ± 0.009 rad. The object to be measured is a rectangular block with a height of approximately 0.660 cm and a length of approximately 80 cm, as shown in Fig. 3
. The measurement of the 3D shape is conducted by using both the ordinary four-step phase shift method and the proposed novel four-step phase shift method.

First, the object is measured using the ordinary four-step phase shift method without correcting the grating cycle. The average fringe cycle is 3.200 ± 0.010 cm, and the fringe cycle varies approximately from 2.8 to 3.8 cm from left to right. The experimental results are shown in Figs. 4 to 6 .

Second, the object is measured using the novel four-step phase shift method with the grating cycle corrected. The corrected fringe cycle is set to 2.860 cm. Experimental results are shown in Figs. 7 to 9 .

Based on the calculation results, the resolution of the proposed method is significantly increased. We now compare the proposed method with the other mainstream approaches, such as the geometric parameter measuring method, the look-up-table (LUT) approach, and the least-squares method [21]. The geometric parameter measuring method is widely used. However, precise determination of several geometric parameters, such as angle, is difficult to achieve in this method, which is its main disadvantage. The LUT approach is useful in system calibration. It requires precise control of the height during the calibration procedure to provide every pixel the relationship between its phase values and dimensions. However, this method is inconvenient and time consuming. The least-squares method is frequently used to determine the phase-to-height relationship. As is often the case with a certain degree of approximation, measurement accuracy is unsatisfactory, and the calculation process is complicated. On the other hand, the proposed method achieves a uniform cycle grating on the reference surface. The advantages are obvious. It is very convenient, simple, practical, and has high accuracy. Complex mathematical computations are not needed. The resulting grating image is suitable for analysis under the Fourier transform profilometry because of its uniform cycle.

## 4. Discussion

Further derivation is performed to analyze and compare the experimental results. For the ordinary four-step phase shift method, the fringe cycle is uniform on the surface *X*′*.* The fringe cycle is calculated using Eq. (1): $\phi (x)=2\pi fLx{({L}^{2}+{d}^{2})}^{1/2}/({L}^{2}+{d}^{2}+d\cdot x)$, and${\omega}_{x}=d\phi (x)/dx=2\pi fL{({L}^{2}+{d}^{2})}^{3/2}/{({L}^{2}+{d}^{2}+d\cdot x)}^{2}$.Thus, ${T}_{x}=T{({L}^{2}+{d}^{2}+d\cdot x)}^{2}/[L{({L}^{2}+{d}^{2})}^{3/2}]$, where *T* is the fringe cycle of the surface *X*′, it has a value of 2.860 cm, and *T _{x}* is the fringe cycle of the surface

*X*. The relationship curve of

*T*and

_{x}*x*is shown in Fig. 10 .

For the ordinary four-step phase shift method, the phases on the projection and reference surfaces are not strictly consistent, causing height error. Nevertheless, the experiment shows that the height error is slight. Thus, the error caused by cycle broadening is further investigated.

The actual height of the object is approximately 0.660 cm. It is known that $h=L\Delta \varphi (x)/[2\pi {f}_{x}d+\Delta \varphi (x)]$. The height error of the mathematical simulation from the ordinary four-step phase shift method is shown in Table 1
, where $\overline{{T}_{x}}$ is the average fringe cycle of the surface *X*. $\overline{{T}_{x}}$ is 3.200 ± 0.010 cm, and $\overline{{f}_{x}}$ is 0.313 ± 0.001 cm^{−1}. $\overline{{h}_{x}}$ is the height of the object measured using the ordinary four-step phase shift method, and ${h}_{x}$ is the height obtained using the grating with changing frequency. ${h}_{x}$ is consistent with the actual height of the object. $\Delta \varphi (x)$ is the calculated phase difference in the position of *x*. $\Delta {h}_{x}=\overline{{h}_{x}}-{h}_{x}$ is the height error.

Table 1 shows that the height of the object measured through the ordinary four-step phase shift method decreases as *x* increases. When $\overline{{T}_{x}}$>${T}_{x}$, the measured height is greater than the actual height of the object, whereas when $\overline{{T}_{x}}$<${T}_{x}$, the measured height is smaller than the actual height. The simulation results are consistent with those shown in Fig. 6(d). When the pixels are increased from 300 to 1200, the measured height of the object is decreased. When the pixels are increased from 300 to 600, the measured height of the object becomes greater than the actual height. In contrast, the measured height of the object becomes significantly smaller than the actual height as the pixels are increased from 800 to 1200. When the pixels are increased from 600 to 800, the difference between the measured height and the actual height of the object becomes smaller. As *x* increases, the height of the object measured through the ordinary four-step phase shift method becomes smaller than the actual height. Moreover, the error also increases. At 1000 pixels, the error is approximately 0.1 cm, and the relative error is 15.2%. These results are not acceptable in high-precision measurements. Therefore, the ordinary fringe projection profilometry is suitable only for measuring small objects.

When using the novel four-step phase shift method, the fringe cycle on the reference surface is set at a fixed value. This resolves the inconsistency of the phase shift on the projection and reference surfaces. As shown in Fig. 9(d), the 3D shape measurement results obtained through the proposed method is 0.662 ± 0.008 cm. The measured height is constant and consistent with the actual height. Moreover, height error is significantly reduced compared with that of the traditional method, as seen in Fig. 6(d). Therefore, the measurement accuracy of the proposed method is considerably higher than that of the ordinary method. At 1000 pixels, the error is smaller than 0.01 cm, and the relative error is less than 1.5%. Measurement accuracy is increased approximately 10 times.

Because both sides of the object are clamped, the height of the middle part of the object is higher than those of both sides. The height of the object when the pixels are increased from 400 to 600 is slightly higher than those of both sides, as seen in Figs. 6(d) and 9(d).

## 5. Conclusion

Fringe projection profilometry based on a novel phase shift method can correct cycle error and also significantly improve measurement accuracy. The proposed method can be widely used in measuring the 3D shapes of objects, particularly those of large objects.

## Acknowledgments

This work was supported by the Natural Science of Jiangxi Province of China and the Jiangxi Education Department.

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