Computer-generated Moir&#x00E9; profilometry

Chengmeng Li; Yiping Cao; Cheng Chen; Yingying Wan; Guangkai Fu; Yapin Wang

doi:10.1364/OE.25.026815

1. Introduction

For optical three-dimension (3D) measurement technique [1–6] has many advantages such as noncontact, high resolution, high precision, nondestructive, high-speed data acquisition, wide measurement range, easy to realize automation and so on, it is widely used in 3D shape measurements. In order to satisfy the growing demands of real-time measurements, an increasing number of single-shot 3D measurement techniques [7-8] attract scholars’ attention.

The single frame 3D measurement techniques meet the requirements of real-time measurement well since only one deformed pattern is required to get the surface information of the object. Fourier transform profilometry (FTP) was proposed by Takeda M and Mutoh K [9]. in 1983 and developed maturely with the efforts of many researchers [10–12]. FTP can get the 3D shape of the object by using only one fringe pattern on the reference and one deformed pattern on the object thus it has advantages such as single frame acquisition, full field analysis, high resolution and so on. But the filtering behavior in the frequency domain causes frequency leakage, spectrum aliasing and other problems so that the measurement range is limited and some details may be lost. Zhang and Pan et al. [13–15] proposed a color-encoded fringe projection phase shifting technique which uses R, G and B channels to achieve three steps of phase-shifting. However, it has problems about color coupling caused by overlapping spectra between neighboring channels and the gray imbalance caused by camera’s different sensitivity to three colors. Hence, being sensitive to color is a weakness which may introduce inevitable errors easily in phase calculation. Guan et al. [16] proposed a method based on composite grating projection, and aiming at the problem about spectrum overlap in the method, He et al. [17] proposed a method with orthogonal composite grating aided by fringe contrast and background calibration. In this kind of method, each phase-shifted sinusoidal grating is respectively modulated into corresponding cosine carrier wave with distinct frequencies, and all the modulated carrier waves are summed together to be as illumination. This method sums more than three fringes together to share 255 gray levels, so each fringe has a small gray range, which leads a lower measurement precision and range. Besides, Zhu et al. [18, 19] proposed a gray scale imbalance correction in real-time phase measuring profilometry (RPMP), which effectively solves the problems about color coupling by using color wheels to project R, G and B three gratings in sequence and controlling CCD to capture deformed fringes synchronously. But the problem of the RPMP is somehow limited by the refresh rate of the projector. In moiré profilometry [20, 21], when two high frequency fringe patterns are superimposed together, the product will contain the moiré fringe as its low-frequency component. And the phase information only modulated by the object is available in moiré fringes. Recently, a high speed moiré based phase retrieval method [22] has been presented for thin objects. Different from traditional moiré profilometry, the moiré fringes are generated by computer in this method. However, arccosine computing is used to calculate the phase so that the phase distribution is required to be within (0, $π$ ]. Once the phase modulated by the object is greater than $π$ it will go wrong in phase calculation. What’s more, the influence of reflectance to fringe contrast is not taken into account, so fringe contrast is still an unknown factor which has impacts on phase calculation. So a new real-time measurement method based on computer-generated moiré profilometry (CGMP) is proposed in this paper. Only a single-shot deformed pattern is required to retrieve the phase distribution of the object, and the obtained moiré fringes are generated by computer rather than moiré fringe direct acquisition. It cannot only effectively avoid the influence of the transient caused by moiré fringe direct acquisition, but also be insensitive to color or texture and inhomogeneous light field.

2. Principle of CGMP

Different from the traditional moiré profilometry in which moiré fringe is captured directly, the optical setup of CGMP is just the same as that of PMP or FTP as shown in Fig. 1.

Fig. 1 Optical measurement system.

Download Full Size | PDF

First of all, the previous preparation must be accomplished. Four sinusoidal gratings, with a shifted phase of $π / 2$ , are projected to the reference plane. And the four corresponding captured fringe patterns can be expressed as:

I_{1} (x, y) = R (x, y) {A + B \cos [2 π f_{0} x + φ_{0} (x, y)]},

I_{2} (x, y) = R (x, y) {A + B \cos [2 π f_{0} x + φ_{0} (x, y) + π / 2]},

I_{3} (x, y) = R (x, y) {A + B \cos [2 π f_{0} x + φ_{0} (x, y) + π]},

I_{4} (x, y) = R (x, y) {A + B \cos [2 π f_{0} x + φ_{0} (x, y)] + 3 π / 2} .

Where

R (x, y)

denotes reflectance coefficient of the reference plane,

A

and

B

respectively present the background light intensity and the fringe contrast, and

φ_{0} (x, y)

is the phase distribution modulated by the reference plane. In order to get the AC component of the fringe pattern

I_{1} (x, y)

, we can use a 180-degree phase shifting on the reference plane in advance to eliminate

R (x, y) A

exactly rather than calculate the phase on the reference plane

φ_{0} (x, y)

. In the same way, a 270-degree phase-shifted fringe pattern can be used to get another AC component of the fringe pattern

I_{2} (x, y)

. Then, the AC components of the 0-degree and 90-degree phase-shifted fringe patterns can be expressed as Eq. (5) and (6) respectively:

{\tilde{I}}_{0}^{R} (x, y) = \frac{1}{2} [I_{1} (x, y) - I_{3} (x, y)] = R (x, y) B \cos [2 π f_{0} x + φ_{0} (x, y)] .

{\tilde{I}}_{90}^{R} (x, y) = \frac{1}{2} [I_{2} (x, y) - I_{4} (x, y)] = R (x, y) B \cos [2 π f_{0} x + φ_{0} (x, y) + π / 2] .

So the above two AC components are refined extracted and saved in computer in advance.

While measuring, only one sinusoidal grating needs to be projected onto the measured object and only the one corresponding deformed pattern needs to be captured.

\begin{array}{l} I_{o} (x, y) = R^{'} (x, y) {A + B \cos [2 π f_{0} x + φ (x, y)]} \\ = R^{'} (x, y) A + R^{'} (x, y) B \cos [2 π f_{0} x + φ (x, y)] . \end{array}

Where

R^{'} (x, y)

denotes reflectance coefficient of the object surface, and

φ (x, y)

is the phase modulated by both the measured object and the reference plane. Transforming the deformed pattern into the frequency domain and filtering out the DC component, the AC component remained can be expressed as:

\begin{array}{l} {\tilde{I}}_{0}^{O} (x, y) = R^{'} (x, y) B \cos [2 π f_{0} x + φ (x, y)] \\ = I_{o} (x, y) - a b s (F F T^{- 1} {F F T {I_{o} (x, y)} r e c t (x / f_{x \max}^{O}, y / f_{y \max}^{O})}) . \end{array}

Where

f_{x \max}^{O}, ​ f_{y \max}^{O}

are the width and the height of the filter window. While this AC component

{\tilde{I}}_{0}^{O} (x, y)

is multiplied by the two saved AC components respectively, the obtained results can be expressed by Eq. (9) and (10).

\begin{array}{l} I_{0}^{O R} (x, y) = {\tilde{I}}_{0}^{O} (x, y) \times {\tilde{I}}_{0}^{R} (x, y) = \frac{1}{2} R (x, y) R^{'} (x, y) B^{2} \cos [4 π f_{0} x + φ_{0} (x, y) + φ (x, y)], \\ + \frac{1}{2} R (x, y) R^{'} (x, y) B^{2} \cos [φ (x, y) - φ_{0} (x, y)] \end{array}

\begin{array}{l} I_{90}^{O R} (x, y) = {\tilde{I}}_{0}^{O} (x, y) \times {\tilde{I}}_{0}^{R} (x, y) = - \frac{1}{2} R (x, y) R^{'} (x, y) B^{2} \sin [4 π f_{0} x + φ_{0} (x, y) + φ (x, y)] . \\ + \frac{1}{2} R (x, y) R^{'} (x, y) B^{2} \sin [φ (x, y) - φ_{0} (x, y)] \end{array}

It is obvious that the second item in Eq. (9) or (10) is just what we called the computer-generated moiré fringe. It contains the phase information of the measured object with the phase of reference plane eliminated though it also contains the contrast of the projected fringe and the reflectance coefficients of both the object surface and the reference plane. We define the moiré fringe in Eq. (9) and (10) as 0-degree and 90-degree moiré fringe respectively. They can be extracted by just using a low-pass filter in frequency domain and the results are shown as Eq. (11) and (12).

\begin{array}{l} I_{m o i r e 0} (x, y) = \frac{1}{2} R (x, y) R^{'} (x, y) B^{2} \cos [φ (x, y) - φ_{0} (x, y)], \\ = F F T^{- 1} {F F T {I_{0}^{O R} (x, y)} r e c t (x / f_{x \max}^{m}, y / f_{y \max}^{m})} \end{array}

\begin{array}{l} I_{m o i r e 90} (x, y) = \frac{1}{2} R (x, y) R^{'} (x, y) B^{2} \sin [φ (x, y) - φ_{0} (x, y)] \\ = F F T^{- 1} {F F T {I_{90}^{O R} (x, y)} r e c t (x / f_{x \max}^{m}, y / f_{y \max}^{m})} . \end{array}

Where $f_{x \max}^{m}, f_{y \max}^{m}$ are the width and the height of the filter window for moiré fringe extraction. The tangent of the phase only modulated by the height of the object can be obtained by Eq. (13) with the influences of the contrast of the projected grating and the reflectance coefficient of both the object surface and the reference plane eliminated.

\tan [φ (x, y) - φ_{0} (x, y)] = \frac{I_{m o i r e 90} (x, y)}{I_{m o i r e 0} (x, y)} .

So is

φ (x, y) - φ_{0} (x, y)

wrapped within (

- π, π

] due to arctangent calculation. The phase unwrapping method [23, 24] is used to transform this phase into unwrapped phase

ϕ (x, y)

. The mapping relation between the phase and the height can be obtained by calibration [25].

\frac{1}{h (x, y)} = a (x, y) + b (x, y) \frac{1}{ϕ (x, y)} + c (x, y) \frac{1}{ϕ^{2} (x, y)} .

Where

a (x, y)

,

b (x, y)

and

c (x, y)

are the parameters obtained by calibration. So the object can be reconstructed by this mapping relationship. The process flow chart is shown in Fig. 2. The dot line part shows the pre-preparation process while the dash line part shows the measuring process.

Fig. 2 The process flow chart of computer generated moiré profilometry.

Download Full Size | PDF

3. Experimental results and discussions

In order to verify the feasibility of CGMP, both simulations and experiments are carried out.

3.1 Simulations

Peaks function shown in Fig. 3(a) is used to be as the measured object model. In order to show the anti-noise capability of CGMP, two percent random noise is introduced in both simulated fringe patterns and the simulated deformed pattern. Just as shown in Fig. 2, four fringe patterns with a shifted phase of $π / 2$ on the reference plane must be prepared and the two AC components of 0-degree and 90-degree phase-shifted fringe patterns can be refined calculated and saved in computer before measuring. When measuring, only a single-shot deformed pattern modulated by measured object as shown in Fig. 3(b) is required. The computer-generated moiré fringes at 0 degree and 90 degree as shown in Fig. 3(c) and 3(d) can be extracted respectively. The wrapped phase modulated by the height of the object can be obtained as shown in Fig. 3(e). After phase unwrapping and phase-to-height mapping, the 3D shape of the object can be reconstructed successfully as shown in Fig. 3(f).

Fig. 3 Stimulated results: (a) simulated object; (b) the deformed pattern; (c) 0-degree moiré fringe; (d) 90-degree moiré fringe; (e) wrapped phase; (f) reconstructed object.

Download Full Size | PDF

Furthermore, CGMP has the single-shot projection feature as FTP and both two methods can obtain object’s phase information directly with no requirement for reference plane’s phase in advance. So a further comparison simulation about the error distributions with FTP and CGMP are carried out as shown in Fig. 4. Figure 4(a) shows the error distributions with FTP while Fig. 4(b) shows the error distribution with CGMP. It is intuitive that FTP has a larger error at some areas where height changes quickly, while the error of the proposed method is nearly equally distributed in a narrow range. It adequately shows the validity of 3D measurement based on computer-generated moiré profilometry with a higher measuring accuracy and repeatability.

Fig. 4 Error distributions: (a) with FTP; (b) with CGPM.

Download Full Size | PDF

3.2 Experimental results

In order to verify the validity and practicality of CGMP, many experiments for different complex measured objects have been done. The setup as Fig. 1 is used. The type of the DLP is View Sonic PLED-W200 and the monochrome CCD camera is MVC1000MF.

The measured object is a heart model as shown in Fig. 5(a). In the pre-preparation process, four fringe patterns with a shifted phase of $π / 2$ on the reference plane are captured by CCD camera, and the two AC components of 0-degree and 90-degree phase-shifted fringe patterns are refined calculated with Eq. (5) and (6) and saved in computer in advance. Then in the measuring process, the object is put on the reference plane and the grating is projected onto the object. The captured deformed pattern is shown in Fig. 5(b). The 0-degree moiré fringe and 90-degree moiré fringe are obtained with CGMP as shown in Fig. 5(c) and 5(d). Then the wrapped phase modulated by the height of the measured object can be obtained, just as shown in Fig. 5(e). After phase unwrapping and phase-to-height mapping, the heart model is effectively reconstructed as shown in Fig. 5(f).

Fig. 5 Measuring results of CGMP: (a) measured object; (b) deformed pattern; (c) 0-degree moiré fringe; (d) 90-degree moiré fringe; (e) wrapped phase; (f) reconstructed object.

Download Full Size | PDF

Further more, a face mask with some color painted as shown in Fig. 6(a) is also measured to verify its ability to avoid the interference of color or texture. Figure 6(b) is the captured deformed pattern, Fig. 6(c) is the wrapped phase of the object and Fig. 6(d) is the reconstructed object based on CGMP. For a better observation, the cutaway views at row 420 (solid line in Fig. 6(b)) and 535 (dash-dot-dot line in Fig. 6(b)) where some painted color is included are shown in Fig. 6(e) and 6(f) respectively. Clearly, the surfaces are almost the same smooth as those with no color painted. It is obvious that the CGMP has an ability to effectively avoid the interference of color or texture. This feature can also be revealed from the theoretical analysis. Because both the extracted computer-generated 0-degree moiré fringe and 90-degree moiré fringe have the same factor $R (x, y) R^{'} (x, y)$ which reflects the color or texture characteristics of the reference plane and the measured object as expressed as Eq. (11) and (12), the ratio of these two computer-generated moiré fringes can remove this factor by reduction as long as the factor is not zero. And this ratio just reflects $\tan [φ (x, y) - φ_{0} (x, y)]$ which is definitely independent of the reflectance factors.

Fig. 6 Measuring results of a colored object: (a) the measured object with color painted; (b) the deformed pattern; (c) wrapped phase of the object; (d) reconstructed object; (e) cutaway views at row 420; (f) cutaway views at row 535.

Download Full Size | PDF

Figure 7 shows the comparison experimental results between FTP and CGMP. As shown in Fig. 7(a), the measured object is a duotone eraser with steep edges. The captured deformed pattern is shown in Fig. 7(b). The cutaway views at column 420 of the reconstruction results with PMP, FTP, and CGMP respectively are shown in Fig. 7(c). As is known to all, PMP is of the highest accuracy. Its result is taken as the quasi truth-value. Obviously the result of CGMP is closer to the quasi truth-value than that of the FTP. The reconstructed edges with CGMP remain steep while that with FTP is smoothed. Meanwhile, the result at dark color with CGMP is still close to the quasi truth-value while that with FTP is deviated from its truth-value. So CGMP has the merit of being more insensitive to color or texture. That is, CGMP is of great potential in real-time measurement for its single-shot feature just like FTP but owns a higher accuracy than FTP. The higher accuracy feature can also be revealed from frequency domain analysis.

Fig. 7 Comparison between FTP and CGMP for measuring the steep object: (a) the measured object; (b) the deformed pattern; (c) the cutaway views at column 450 with PMP, FTP and CGMP.

Download Full Size | PDF

Figure 8(a) and 8(b) show the spectrum distributions with FTP and CGMP respectively. In FTP, the phase information is obtained by extracting the fundamental component of the deformed pattern, so both the zero component and high frequency component have impacts on its measuring result which make it difficult to determine the filtering window. A wide filter will introduce a lot of noise; on the contrary, a narrow one will cause a loss of information. But in the CGMP, the zero components are removed before generating superimposed products so that the background’s impact to measuring result can be greatly reduced. Besides, the fundamental frequency with CGMP is twice that with FTP, the separated distance between the zero frequency and fundamental frequency with CGMP is much father than that with FTP. So the probability caused by the spectrum overlapping between the zero frequency and fundamental frequency will be less than that with FTP. In addition, the filter window in CGMP can be wilder than that in FTP due to its higher first-order spectrum, so high frequency information is more abundant in CGMP which will lead a high-fidelity reconstructed 3D shape. To be sure, the error with CGMP is smaller than that with FTP.

Fig. 8 Spectrum distributions in different conditions: (a) with FTP; (b) with CGMP.

Download Full Size | PDF

To further verify the accuracy and repeatability of CGMP, a series of planes with known height are measured. During measurement, the planes with height of 6mm, 10mm, 15mm, and 23mm are measured with FTP and CGMP respectively, and the results are shown in Table 1. Where $h$ indicates the known height of the planes and $\bar{h}$ indicates the average height of the reconstructed plane. Meanwhile, the mean absolute error (MAE) is used to express the accuracy of measurement while the root mean square error (RMS) is used to express the repeatability of measurement. It is clearly found that the proposed method has higher measurement precision and repeatable accuracy compared to FTP.

Table 1. Experimental results for different known heights (/ mm).

View Table

4. Conclusions

A 3D measurement method based on computer-generated moiré profilometry is proposed in this paper. It has the advantages of avoiding the interference of color or texture because the phase calculation of the proposed method is independent of the surface reflectance of both the reference plane and the measured object as long as the reflectance is not zero. As is well-known, FTP is a recognized real-time measurement technique since only one deformed pattern is required to reconstruct the object. The proposed computer-generated moiré profilometry also requires only one deformed pattern while measuring. So it can meet the needs of real-time or online measurement. Simultaneously, the experimental results prove that it has a better measuring accuracy than FTP.

Funding

863 National Plan Foundation of China (2007AA01Z333); Special Grand National Project of China (2009ZX02204-008).

Acknowledgements

863 National Plan Foundation of China; Special Grand National Project of China.

References and links

1. J. Geng, “Structured-light 3D surface imaging: a tutorial,” Adv. Opt. Photonics 3(2), 128–160 (2011).

2. V. Srinivasan, H. C. Liu, and M. Halioua, “Automated phase-measuring profilometry of 3-D diffuse objects,” Appl. Opt. 23(18), 3105 (1984). [PubMed]

3. C. Jiang, S. Jia, J. Dong, Q. Bao, J. Yang, Q. Lian, and D. Li, “Multi-frequency color-marked fringe projection profilometry for fast 3D shape measurement of complex objects,” Opt. Express 23(19), 24152–24162 (2015). [PubMed]

4. K. Peng, Y. Cao, Y. Wu, and M. Lu, “A new method using orthogonal two-frequency grating in online 3D measurement,” Opt. Laser Technol. 83, 81–88 (2016).

5. Y. Wan, Y. Cao, C. Chen, and K. Peng, “An online triple-frequency color-encoded fringe projection profilometry for discontinuous object,” J. Mod. Opt. 63(14), 1–8 (2016).

6. K. Zhong, Z. Li, R. Li, Y. Shi, and C. Wang, “Pre-calibration-free 3D shape measurement method based on fringe projection,” Opt. Express 24(13), 14196–14207 (2016). [PubMed]

7. C. Jiang, T. Bell, and S. Zhang, “High dynamic range real-time 3D shape measurement,” Opt. Express 24(7), 7337–7346 (2016). [PubMed]

8. S. V. D. Jeught and J. J. J. Dirckx, “Real-time structured light profilometry: a review,” Opt. Lasers Eng. 87, 18–31 (2016).

9. M. Takeda and K. Mutoh, “Fourier transform profilometry for the automatic measurement of 3-D object shapes,” Appl. Opt. 22(24), 3977–3982 (1983). [PubMed]

10. X. Su, W. Chen, Q. Zhang, and Y. Cao, “Dynamic 3-D shape measurement method based on FTP,” Opt. Lasers Eng. 36(1), 49–64 (2001).

11. X. Mao, W. Chen, and X. Su, “Improved Fourier-transform profilometry,” Appl. Opt. 46(5), 664–668 (2007). [PubMed]

12. J. F. Casco-Vasquez, R. Juarez-Salazar, C. Robledo-Sanchez, G. Rodriguez-Zurita, F. G. Sanchez, L. M. A. Aguilar, and C. Meneses, “Fourier normalized-fringe analysis by zero-order spectrum suppression using a parameter estimation approach,” Opt. Eng. 52(7), 074109 (2013).

13. S. Zhang and P. S. Huang, “High-resolution, real-time three-dimensional shape measurement,” Opt. Eng. 45(12), 1269–1278 (2006).

14. S. Zhang, “Recent progresses on real-time 3D shape measurement using digital fringe projection techniques,” Opt. Lasers Eng. 48(2), 149–158 (2010).

15. J. Pan, P. S. Huang, and F. P. Chiang, “Color phase-shifting technique for three-dimensional shape measurement,” Opt. Eng. 45(1), 013602 (2006).

16. C. Guan, L. G. Hassebrook, D. L. Lau, D. L. Lau, V. G. Yalla, and C. J. Casey, “Improved composite-pattern structured-light profilometry by means of post-processing,” Opt. Eng. 47(47), 7203 (2008).

17. Y. He and Y. Cao, “Three-dimensional measurement method with orthogonal composite grating aided by fringe contrast and background calibration,” Opt. Eng. 49(7), 717–720 (2010).

18. L. Zhu, Y. Cao, D. He, and C. Chen, “Gray scale imbalance correction in real-time phase measuring profilometry,” Opt. Commun. 376, 72–80 (2016).

19. L. Zhu, Y. Cao, D. He, and C. Chen, “Real-time tricolor phase measuring profilometry based on CCD sensitivity calibration,” J. Mod. Opt. 64(4), 379–387 (2017).

20. J. J. J. Dirckx, W. F. Decraemer, and G. Dielis, “Phase shift method based on object translation for full field automatic 3-D surface reconstruction from moire topograms,” Appl. Opt. 27(6), 1164–1169 (1988). [PubMed]

21. W.-Y. Chang, F.-H. Hsu, K.-H. Chen, J.-H. Chen, and K. Y. Hsu, “Heterodyne moiré surface profilometry,” Opt. Express 22(3), 2845–2852 (2014). [PubMed]

22. S. Wang, K. Yan, and L. Xue, “High speed moiré based phase retrieval method for quantitative phase imaging of thin objects without phase unwrapping or aberration compensation,” Opt. Commun. 359, 272–278 (2016).

23. X. Su and L. Xue, “Phase unwrapping algorithm based on fringe frequency analysis in Fourier-transform profilometry,” Opt. Eng. 40(4), 637–643 (2001).

24. X. Su and W. Chen, “Reliability-guided phase unwrapping algorithm: a review,” Opt. Lasers Eng. 42(3), 245–261 (2004).

25. X. Su, W. Song, Y. P. Cao, and L. Xiang, “Both phase height mapping and coordinates calibration in PMP,” Proc. SPIE 4839, 874–875 (2002).

$h$	6		10		15		23
Method	FTP	CGMP	FTP	CGMP	FTP	CGMP	FTP	CGMP
$\bar{h}$	6.016	5.991	10.029	10.007	15.005	14.985	22.999	22.991
MAE	0.222	0.220	0.164	0.153	0.128	0.102	0.117	0.097
RMS	0.272	0.270	0.200	0.188	0.162	0.124	0.149	0.122

Computer-generated Moiré profilometry

Abstract

1. Introduction

2. Principle of CGMP

3. Experimental results and discussions

3.1 Simulations

3.2 Experimental results

4. Conclusions

Funding

Acknowledgements

References and links

Cited By

Figures (8)

Tables (1)

Equations (14)

Optics Express