Instability of projection light source and real-time phase error correction method for phase-shifting profilometry

Cheng Chen; Yingying Wan; Yiping Cao

doi:10.1364/OE.26.004258

1. Introduction

Among the existing three-dimensional (3D) shape measurement techniques, the phase-shifting profilometry (PSP) is one of the most commercialized and effective methods due to its low cost, high accuracy and high reliability [1–4]. It has been widely used in virtual reality [5], reverse engineering [6], bio-medical engineering [7], and plastic surgery [8] and so on. In PSP, a series of sinusoidal fringe patterns are usually adopted to obtain the phase information which is used for recovering the 3D geometry of the measured object. However, its measurement accuracy is usually affected by such error sources as the phase shift [9], the nonlinearity of the measurement system [10,11], camera noise [12] and system vibration [13,14].

In high-accuracy measurements, all of these error sources mentioned above cannot be ignored. With the advent of digital projectors and some new projection devices [15], the phase-shift error has become negligible. In contrast, the nonlinearity of the measurement system becomes more serious even using cameras with linear response, for the reason that digital projectors are usually nonlinear. Consequently, a multitude of techniques have been proposed to reduce the nonlinear errors, such as phase error compensation methods [16,17], gamma correction methods [18,19] and a large-step phase-shifting method [20]. The measurement accuracy will be significantly improved after reducing the nonlinear errors. However, the camera noise and system vibration will lead that the intensity of captured fringe patterns deviate from the true value at different time, which could not represent the real situation. As a consequence, phase errors will be introduced in the next phase-shifting calculations using these fringe patterns. To address this problem, multi-frame average algorithm was adopted by Yao et al [13], and it worked well for his purpose. But, there are some errors still cannot be completely eliminated even with a large numbers of images in practical measurement. Besides of the camera noise and system vibration, the fluctuations of the captured fringe patterns may also be caused by other factors. Several solutions have been studied for its improvements. Zhang et al. proposed a method based on empirical mode decomposition algorithm to eliminate the intensity fluctuations caused by background light and the intensity of projection system light source [21]. Lu et al. improved this problem by using an intensity stabilized xenon lamp as the light source [22]. While the above two methods have worked well, they either increase algorithm complexity or hardware aid [23]. Xu et al. developed a simple averaging algorithm for correction by extracting the relative instant intensity coefficient of each interferogram [24]. Lu et al. corrected the effect of the illumination fluctuations based on the histograms of fringe patterns with efficient computation [23]. These two methods were effective, yet their performance of the correction both depend on the number of fringes in fringe patterns, which will restrict their applications to some extent. Furthermore, by reviewing above mentioned techniques, the characteristics of the variation of the light source in PSP with digital projectors have not been discussed. And a rigorous mathematical model for the phase error induced by the instability of projection light source (IPLS) has not yet been unveiled, to the best of our knowledge.

In this study, firstly, we investigate the characteristics of IPLS using a specially designed experimental setup, and find that IPLS will cause notable random variation with time in the intensity of the captured patterns while its variation rate is equivalent in the full field of view at the same moment. Then, based on the results of the investigation, a new mathematical model to analyze the phase error considering the effect of IPLS is proposed for PSP. Within the model, the induced phase error is derived, and it is verified by experiments that a mapping function between the phase error and wrapped phase can be established in a measurement period. However, because IPLS causes the intensity to vary randomly with time, the time-dependent phase error hardly can be uniquely determined by a function for different measurements. As a result, pre-calibration procedures may fail to reduce the induced time-dependent phase error which requires to be corrected in real time. Finally, for the first time, we propose two real-time correction methods for the time-dependent phase error induced by IPLS using a new designed 3D shape measurement system. The feasibility and validity of the two proposed correction methods are verified by experiments.

This paper is organized as follows: In Section 2, IPLS of the projection devices is investigated, and a new mathematical model to analyze the time-dependent phase error induced by IPLS in PSP is proposed and verified. In Section 3, two real-time correction methods for IPLS are proposed and examined by experimental results. Section 4 presents the discussions and Section 5 concludes the paper.

2. Principle for error induced by the instability of the projection light source

2.1 Instability of projection light source during experimentation

As we know the instability of the projection light source (IPLS) is a common phenomenon in projection devices. To study the effect of IPLS on the intensity of captured patterns, we investigated the projection light source of a liquid crystal display (LCD) projector (HCP-70X). Besides of the projector, the test experimental setup is specially designed including two different digital cameras (camera 1: GEV-B1610M and camera 2: MVC1000MF) and a flat board, as shown in Fig. 1. The projector was fixed and its LCD control bus interface had been removed to ensure that the variation of the output light intensity was only determined by the projection light source. When the light source was projected on the flat board, the two cameras captured the images of the flat board simultaneously at the speed of 2 fps. The entire experiment was conducted in a dark environment to avoid the interference of the ambient intensity and it lasted more than 1.5 hours.

Fig. 1 Experimental setup.

Download Full Size | PDF

To provide appropriate grayscales for two cameras, a checkerboard pattern with rich grayscales was printed on the flat board and two subregions of the checkerboard shown in Fig. 2(a) were analyzed. The average intensity within a selected region represents the intensity of this region and the intensities of the two subregions in each camera are plotted in Fig. 2(b). From this figure, we can see that IPLS causes the intensities vary randomly with time. They may change slowly in a long period of time while may also change sharply in a short time. Thus, the multi-frame average algorithm adopted in [13] seems not applicable when considering IPLS. However, despite of the strong random variation during the measurement, the four intensity curves have a similar variation trend. We define the intensity variation rate as the ratio of the measured intensity to the ideal intensity, and the variation rates of the four curves are almost equivalent at the same moment within the range of [-3%, 3%] (here considering the mean of the curve as the ideal intensity). Measurements of other types of projectors showed similar behavior. These results demonstrate that the notable random variation in the intensity of the captured patterns is only caused by IPLS of projection devices and is independent to the cameras. Thus, the intensity variation of the captured pattern directly reveals the intensity variation of the projection light source. Meanwhile, the variation rate of the intensity in the full field of view can be regarded as equivalent at the same moment.

Fig. 2 (a) Captured checkerboard pattern, (b) the intensity variation of the regions in (a).

Download Full Size | PDF

2.2 Mathematical model for time-dependent phase error induced by IPLS

The models of PSP used in previous work [18–20] did not take account of the intensity of the projection light source. In this study, as the maximum output intensity of the projected patterns is directly determined by the intensity of the projection light source, we establish the mathematical model considering the effect of IPLS. Assuming the ideal intensity of the light source to be $B^{p} (x, y)$ , without considering the intensity variation, the ideal intensity of the projection fringe pattern $P (x, y)$ can be expressed as

P (x, y) = B^{p} (x, y) [a + b \cos (2 π f x)]

where (x, y) is the coordinate of an arbitrary point in the pattern; f is the spatial frequency of the coded fringes; a and b are user defined constants, and b/a represents the fringe contrast. In a standard N-step (N ≥ 3) phase-shifting technique, the intensity distribution of each captured deformed fringe pattern can be described as

I_{n} (x, y) = R (x, y) {A (x, y) + B^{c} (x, y) {a + b \cos [φ (x, y) + 2 n π / N]}}

where n = 1,2,…,N;

R (x, y)

is the reflective ratio;

A (x, y)

represents the ambient intensity;

B^{c} (x, y)

represents the value of

B^{p} (x, y)

in the captured pattern;

φ (x, y)

is the ideal phase modulated by the height of the measured object. According to the phase function [25], the desired phase

φ (x, y)

can be expressed as

φ (x, y) = arc \tan \frac{\sum_{n = 1}^{N} [I_{n} (x, y) \sin (2 n π / N)]}{\sum_{n = 1}^{N} [I_{n} (x, y) \cos (2 n π / N)]}

However, the experiments presented in Section 2.1 show that the intensity of the projection light source as well as the intensity of the captured patterns will vary with time. Therefore, the computed phase may not be as ideal as given in Eq. (3). To analyze the time-dependent phase error induced by the variation of the intensity, Eq. (1) can be rewritten as

P (x, y) = {L (t) B}^{p} (x, y) [a + b \cos (2 π f x)]

where

L (t)

is the intensity variation rate at the moment of t and it is independent to coordinates, because the variation rate of the full-field intensity is equivalent at the same moment as discussed in Section 2.1. Then, the practical intensity distribution of each deformed fringe pattern can be described as

I_{n}^{L} (x, y) = R (x, y) {A (x, y) + L (t_{n}) B^{c} (x, y) {a + b \cos [φ (x, y) + (2 n π / N)]}}

Where t_n represents the acquisition moment when the nth pattern

I_{n}^{L} (x, y)

was captured, and Eq. (5) can be further expressed as

[I_{n}^{L} (x, y) - R (x, y) A (x, y)] / L (t_{n}) = R (x, y) B^{c} (x, y) {a + b \cos [φ (x, y) + (2 n π / N)]}

Consequently, the desired phase information can be calculated as follow

φ (x, y) = arc \tan \frac{\sum_{n = 1}^{N} {[I_{n}^{L} (x, y) - R (x, y) A (x, y)] / L (t_{n})} \sin (2 n π / N)}{\sum_{n = 1}^{N} {[I_{n}^{L} (x, y) - R (x, y) A (x, y)] / L (t_{n})} \cos (2 n π / N)}

The measured phase

ϕ (x, y)

is usually directly calculated by submitting

I_{n}^{L} (x, y)

into Eq. (3) and it can be considered as the sum of the desired phase and the time-dependent phase error

Δ ϕ (x, y)

induced by IPLS. Thus, we have

ϕ (x, y) = φ (x, y) + Δ ϕ (x, y)

The time-dependent phase error can be derived as the following form

\begin{array}{l} Δ ϕ (x, y) = arc \tan \frac{\sum_{n = 1}^{N} I_{n}^{L} (x, y) \sin (2 n π / N)}{\sum_{n = 1}^{N} I_{n}^{L} (x, y) \cos (2 n π / N)} \\ - arc \tan \frac{\sum_{n = 1}^{N} {[I_{n}^{L} (x, y) - R (x, y) A (x, y)] / L (t_{n})} \sin (2 n π / N)}{\sum_{n = 1}^{N} {[I_{n}^{L} (x, y) - R (x, y) A (x, y)] / L (t_{n})} \cos (2 n π / N)} \end{array}

If the light source operates in a stable state, $L (t_{n}) = 1$ , then $Δ ϕ (x, y) = 0$ , which implies the desired phase information can be directly calculated from the captured fringe patterns. However, due to IPLS, the induced time-dependent phase error is difficult to be correctly obtained by Eq. (9). From Eq. (5) and Eq. (9), it can be seen that the time-dependent phase error is a function of the parameters of $A (x, y)$ , $B^{c} (x, y)$ , $L (t_{n})$ , a, b and $φ (x, y)$ , and it is independent of the frequency of the fringe pattern. Assuming the measurement is performed within a controlled measurement environment that the ambient light and the output intensity of the light source are controlled to be full-field uniform, $A (x, y)$ and $B^{c} (x, y)$ can be approximated as constant coefficients in the full field of view. As a result, the distribution of the time-dependent phase error will only be consistent with that of $φ (x, y)$ and it can be described as

Δ ϕ (x, y) = E [φ (x, y)]

where

E [•]

represents the mapping function between the wrapped phase

φ (x, y)

and the time-dependent phase error

Δ ϕ (x, y)

.

To verify the theoretical derivation, a white flat board was measured. It has been noted that a large-step phase-shifting method can eliminate the nonlinearity of the measurement system [20], therefore, the 16-step phase-shifting algorithm is adopted. Meanwhile, the fringe pitch on the reference plane has been corrected to be uniform by a cycle correction of the projection fringe method [26] in advance to solve the cycle broadening. Then the practical wrapped phase of the flat board is presented as Fig. 3(a). As the wrapped phase for the ideal sinusoidal fringe patterns with horizontal stripes increases linearly from $- π$ to $π$ vertically within one period, the phase error can be computed by taking the difference between the practical wrapped phase and the ideal linear one as shown in Fig. 3(b). The time-dependent phase error in the full field of view is presented as Fig. 3(c) and the phase error corresponding to the practical wrapped phase is further plotted as Fig. 3(d). From Figs. 3(c) and 3(d), we can see that the distribution of the time-dependent phase error induced by IPLS is corresponding to that of wrapped phase, which is in good agreement with the theoretical derivation above.

Fig. 3 (a) Wrapped phase of the flat board. (b) Practical wrapped phase and the ideal wrapped phase. (c) time-dependent phase error distribution of (a). (d) Phase error for all points in (c). The thin curve shows the average of these points.

Download Full Size | PDF

However, since the intensity variation rate $L (t_{n})$ varies with time, it may be unequal in measurements at different times. As a result, the distributions of the time-dependent phase error in each measurement period will be different. Two additional experiments were carried out under the same experimental condition, and the distribution of their time-dependent phase errors is shown in Fig. 4(a) and 4(b), respectively. It can be seen that the distribution of the phase error is random. Therefore, the time-dependent phase error induced by IPLS cannot be uniquely described by just one mapping function so that it cannot be reduced by a pre-calibration procedure. In addition, a simple multi-frame average algorithm also cannot eliminate this time-dependent phase error, which requires to be corrected in real time. In this study, for the first time, we propose two real-time correction methods for the time-dependent phase error induced by IPLS in PSP.

Fig. 4 Time-dependent phase error distribution and phase error curve of (a) the second measurement, (b) the third measurement.

Download Full Size | PDF

3. Correction of IPLS induced time-dependent error

3.1 Phase error correction methods

In contrast to a typical 3D shape measurement system, which consists of a CCD camera, a projector and a flat board as the measurement region, the new designed measurement system is added an extra small flat white board fixed by the side of the measurement region, as shown in Fig. 5. This reference region, captured in the same field of view with the measurement region, is utilized to record the real-time intensity from the projection device for correcting the induced time-dependent phase error.

Fig. 5 Schematic of the new measurement system.

Download Full Size | PDF

In this study, two real-time correction methods are proposed: Firstly, the desired phase can be directly calculated by Eq. (7) using the captured deformed patterns if the parameters of $R (x, y)$ , $A (x, y)$ and $L (t_{n})$ are obtained. Because the intensity variation rate $L (t_{n})$ has been demonstrated to be equivalent in the full field of view at the same moment, the intensity of the captured fringe pattern $I_{n}^{L} (x, y)$ in the measurement region can be corrected using $L (t_{n})$ calculated from the reference region. As shown in Fig. 6(a), the projected pattern on the reference region is divided into two subregions, C1 and C2 is coded with zero and the maximum intensity of the projected fringes on the measurement region (region D), respectively. The first fringe pattern will be chosen as the baseline, and the intensity variation rate $L (t_{n})$ in the reference region can be expressed as

L (t_{n}) = L (t_{1}) (I_{n}^{C 2} - R * A) / (I_{1}^{C 2} - R * A)

where

R * A

is the product of the reflective ratio and the ambient light on the reference region, and can be obtained by averaging the intensity of the region C1;

I_{n}^{C 2}

is the average intensity of nth captured pattern in the region C2. Submitting Eq. (11) into Eq. (7), the desired phase can be derived by

φ (x, y) = arc \tan \frac{\sum_{n = 1}^{N} {[I_{n}^{L} (x, y) - R (x, y) A (x, y)] (I_{1}^{C 2} - R * A) / (I_{n}^{C 2} - R * A)} \sin (2 n π / N)}{\sum_{n = 1}^{N} {[I_{n}^{L} (x, y) - R (x, y) A (x, y)] (I_{1}^{C 2} - R * A) / (I_{n}^{C 2} - R * A)} \cos (2 n π / N)}

In practice the ambient light can be controlled to be uniform and an insignificant level, so the product

R (x, y) A (x, y)

can maintain very small and be approximately equal to a constant in the full field of view, represented by

R * A

. Then, we further rewrite Eq. (12) as

φ (x, y) = arc \tan \frac{\sum_{n = 1}^{N} {[I_{n}^{L} (x, y) - R * A] (I_{1}^{C 2} - R * A) / (I_{n}^{C 2} - R * A)} \sin (2 n π / N)}{\sum_{n = 1}^{N} {[I_{n}^{L} (x, y) - R * A] (I_{1}^{C 2} - R * A) / (I_{n}^{C 2} - R * A)} \cos (2 n π / N)}

This method is named the Real-time Intensity Correction Method (RICM).

Fig. 6 (a) One of the captured patterns of RICM, (b) one of the captured patterns of RPECM.

Download Full Size | PDF

Secondly, analysis and experiments described in Section 2.2 explain and verify that the time-dependent phase error and the wrapped phase satisfy one mapping function $E [•]$ in one measurement period in the full field of view. Therefore, the reference region can be utilized for obtaining the time-dependent phase error and establishing the mapping function $E [•]$ in real time. A look-up table (LUT) that stores the wrapped phase and time-dependent phase errors $(φ (x, y), Δ φ (x, y))$ can be constructed from the fringe patterns in the reference region (region C) shown in Fig. 6(b). To reduce computational cost, the LUT can be constructed using only one period of the fringe patterns and one section of the sinusoidal patterns in the reference region. Then we can use this LUT to compensate the time-dependent phase error of the measurement region. This method is named the Real-time Phase Error Compensation Method (RPECM).

3.2 Experiments

The 3D shape measurement system designed in Section 3.1 is employed to verify the feasibility of the proposed two correction methods. The LCD projector (HCP-70X) with a resolution of 1280 × 960 and the digital camera (GEV-B1610M) with 12bit are employed. The projector has a projection distance of 0.5 m, and the camera is attached with an 8mm focal-length lens and has a native resolution of 1624 × 1236. It should be noted that the reference region is fixed while the measurement region is controlled by a step-motor (SC300-1A) for the calibration of the phase-to-height mapping [27]. Averaging twenty continuously captured patterns was utilized to reduce the camera noise and the effect of system vibration [13]. In order to synchronously evaluate and compare the performance of the two correction methods at the reduction of time-dependent phase errors caused by IPLS, the projected pattern on the reference region is redesigned as shown in Fig. 7(a), region C1 and C2 for RICM, region C for RPECM, respectively. The comparison of root-mean-square (RMS) errors of two methods using different N-step phase-shifting algorithms is presented as Fig. 7(b). It can be seen that the RMS of the time-dependent phase error before correction varies randomly with different N over a wide range within 0.06 radians. The RMS corrected by RICM is as large as that before correction (over 0.04 radians) at N = 3 and 4 because of the nonlinearity effect of measurement system. Whereas, it decreases dramatically to about 0.006 radians at N = 5, followed by some slight fluctuations with increasing N. By contrast, the RMS corrected by RPECM keeps a stable low level with about 0.003 radians within N = 3,4,…,16. From the above experimental results, it should be noticed that RPECM can alleviate the phase error induced by IPLS, and it can also reduce the phase error due to the nonlinearity of the measurement system.

Fig. 7 (a) One of the captured patterns, (b) comparison of RMS of phase errors of two methods using different N-step phase-shifting algorithms.

Download Full Size | PDF

Furthermore, twenty measurements of the plane of the measurement region have been carried out to verify the accuracy and the robustness of the two correction methods. The 16-step phase-shifting algorithm was utilized to avoid the nonlinearity effect. The comparison of phase errors RMS before correction, with RICM and with RPECM in these measurements is presented in Fig. 8(a), where K is the sequence of the measurement. From Fig. 8(a), we can see that the RMS of the time-dependent phase error before correction maintains random large variation in different measurements within [0.0054, 0.0354] radians. In contrast, the RMS of the phase error after both two correction methods are decreased to a stable low level. The mean RMS corrected by RICM and RPECM is 0.0058 and 0.0024 radians, respectively. To give a better illustration of the measurement results, one of the distributions of time-dependent phase error before correction, with RICM and with RPECM is presented in Figs. 8(b)–8(d), respectively. These experimental results demonstrate that the two correction methods can successfully alleviate the time-dependent phase errors induced by IPLS with a good robustness and high accuracy. Moreover, it can be clearly seen that RPECM performs better compared with RICM.

Fig. 8 (a) Comparison of RMS of phase errors using 16-step phase-shifting algorithm with different K. Phase error distribution (when K = 9) (b) before correction, (c) corrected by RICM, and (d) corrected by RPECM.

Download Full Size | PDF

In addition, a leaf vein with complex surface in Fig. 9(a) was measured to verify the validity of the proposed two correction methods. In contrast to the precious measuring system in a small illumination field for small objects [28], the measured leaf vein was measured in a big illumination field (200 × 150 mm²). The 3D shape reconstruction result before correction is shown as Fig. 9(b), which can be clearly seen ripple-like errors on it. The reconstruction results corrected by RICM and by RPECM are shown as Figs. 9(c) and 9(d), respectively. Their reconstruction errors both have been reduced distinctly, whereas, the vein of the leaf is corrected more precisely by RPECM. The details of the leaf vein in the rectangular regions of in Figs. 9(c) and 9(d) are enlarged in Figs. 9(e) and 9(f), respectively. It can be seen that the height range of the vein on the leaf is far less than 1mm and its details have been successfully reconstructed with the proposed two methods in a big illumination field.

Fig. 9 Experimental results of a leaf vein: (a) image of measured object. 3D reconstruction result (b) before correction, (c) corrected by RICM, (d) corrected by RPECM, (e) and (f) enlarged details in the rectangular region of (c) and (d), respectively.

Download Full Size | PDF

Another experiment was conducted on a cup lid to further demonstrate the effectiveness of the proposed methods. Figure 10 present the experimental results and their partially enlarged regions. From Fig. 10, it is clearly seen that the results reconstructed with the proposed two methods are much smoother with well-retained details.

Fig. 10 Experimental results of a cup lid: (a) image of measured object, (b) one of the deformed patterns. 3D reconstruction result (c) before correction, (d) corrected by RICM, and (e) corrected by RPECM, (f-h) enlarged details in the rectangular region of (c-e), respectively.

Download Full Size | PDF

4. Discussion

The instability of projection light source (IPLS) is a common phenomenon in projection device. It may cause notable time-dependent phase error in phase-shifting profilometry (PSP). Therefore, the time-dependent phase error induced by IPLS is necessary to be corrected in real time, especially in high-accuracy measurements. The real-time phase error correction methods proposed in this paper has the following advantages:

• Simple. The system setup is simple since the intensity variation in RICM and the time-dependent phase error in RPECM can be directly obtained from the reference region.
• Robustness. Both of the correction methods correct the time-dependent phase error induced by IPLS in real time, and the experimental results show their robustness.
• Universal. The methods can correct the time-dependent phase errors induced by IPLS for PSP and any systems using a phase-shifting-based method. In addition, the RPECM also has ability to correct the nonlinearity effect of the measurement system.
• Accurate. The experimental results using the two correction methods both achieve high measurement accuracy with RMS of 0.0058 and 0.0024 radians, respectively.

Despite the advantages, the proposed methods have some limitations. The algorithm of RICM is based on the approximation that the product of the reflective ratio and the ambient light is uniform in the full field of view. If the reflective ratio on the measured object is different from that of the reference region, the measurement accuracy may be somewhat reduced. However, because the ambient light can be controlled to be an insignificant level, the product can maintain very small, which will almost have no effect on the measurement accuracy. In RPECM, a LUT is constructed to compensate the phase errors. The measurement accuracy is dependent on the size of the LUT, whereas generation of a large table is somewhat time-consuming. To some extent, we achieve the high measurement accuracy at the cost of sacrificing some measuring speed.

5. Conclusion

The time-dependent phase error induced by the instability of projection light source (IPLS) of projection devices in phase-shifting profilometry (PSP) was systematically studied. IPLS causes notable random variation in the intensity of the captured patterns and results in significant phase error in the measurement. The result from the specially designed experiments indicated that the variation rate of the full-field intensity is equivalent at the same moment. The mathematical model to analyze the time-dependent phase error induced by IPLS for PSP was constructed and verified. It indicates that the induced time-dependent phase error can be approximated as a mapping function with the wrapped phase in one measurement period. However, the function cannot be uniquely described for different measurements as the intensity variation rate changes with time. For the first time, we propose two real-time phase error correction methods using a new designed 3D shape measurement system for IPLS. Experimental results demonstrated the Real-time Intensity Correction Method (RICM) and Real-time Phase Error Compensation Method (RPECM) can achieve high accuracy with RMS of 0.0058 and 0.0024 radians, respectively, and both of them have a good robustness. In addition, RPECM is preferable in practical high-accuracy measurement for its higher accuracy and the ability to eliminate the nonlinearity of the measurement system.

Funding

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

Acknowledgments

This work is supported by 863 National Plan Foundation of China and Special Grand National Project of China.

References and links

1. J. Dai, Y. An, and S. Zhang, “Absolute three-dimensional shape measurement with a known object,” Opt. Express 25(9), 10384–10396 (2017). [CrossRef] [PubMed]

2. J. Zhu, P. Zhou, X. Su, and Z. You, “Accurate and fast 3D surface measurement with temporal-spatial binary encoding structured illumination,” Opt. Express 24(25), 28549–28560 (2016). [CrossRef] [PubMed]

3. Z. Li, K. Zhong, Y. F. Li, X. Zhou, and Y. Shi, “Multiview phase shifting: a full-resolution and high-speed 3D measurement framework for arbitrary shape dynamic objects,” Opt. Lett. 38(9), 1389–1391 (2013). [CrossRef] [PubMed]

4. S. Gorthi and P. Rastogi, “Fringe projection techniques: whither we are?” Opt. Lasers Eng. 48(2), 133–140 (2010). [CrossRef]

5. P. Garbat and M. Kujawińska, “Combining fringe projection method of 3D object monitoring with virtual reality environment:concept and initial results,” in Proceeding of International Symposium on 3d Data Processing (IEEE, 2002), pp. 504–508.

6. S. Caspar, M. Honegger, S. Rinner, P. Lambelet, C. Bach, and A. Ettemeyer, “High speed fringe projection for fast 3D inspection,” Proc. SPIE 8082(1), 57–68 (2011).

7. Y. Wang, J. I. Laughner, I. R. Efimov, and S. Zhang, “3D absolute shape measurement of live rabbit hearts with a superfast two-frequency phase-shifting technique,” Opt. Express 21(5), 5822–5832 (2013). [CrossRef] [PubMed]

8. F. Berryman, P. Pynsent, J. Fairbank, and S. Disney, “A new system for measuring three-dimensional back shape in scoliosis,” Eur. Spine J. 17(5), 663–672 (2008). [CrossRef] [PubMed]

9. P. Hariharan, “Phase-shifting interferometry: minimization of system errors,” Appl. Opt. 39, 967– 969 (2001).

10. S. Ma, C. Quan, R. Zhu, and C. J. Tay, “Investigation of phase error correction for digital sinusoidal phase-shifting fringe projection profilometry,” Opt. Lasers Eng. 50(8), 1107–1118 (2012). [CrossRef]

11. Z. Cai, X. Liu, H. Jiang, D. He, X. Peng, S. Huang, and Z. Zhang, “Flexible phase error compensation based on Hilbert transform in phase shifting profilometry,” Opt. Express 23(19), 25171–25181 (2015). [CrossRef] [PubMed]

12. R. Cai, Q. Wu, W. Shi, H. Shu, Y. Wu, and Z. Wang, “CCD performance model and noise control,” in Proceedings of International Conference on Image Analysis and Signal Processing (IEEE, 2011), pp. 389–394.

13. J. Yao and Y. Zhou, “Phase error elimination considering gamma nonlinearity, system vibration, and noise for fringe projection profilometry,” Opt. Eng. 53(9), 094102 (2014). [CrossRef]

14. L. L. Deck, “Suppressing phase errors from vibration in phase-shifting interferometry,” Appl. Opt. 48(20), 3948–3960 (2009). [CrossRef] [PubMed]

15. M. Fujigaki, T. Sakaguchi, and Y. Murata, “Development of a compact 3D shape measurement unit using the light-source-stepping method,” Opt. Lasers Eng. 85, 9–17 (2016). [CrossRef]

16. S. Zhang, “Comparative study on passive and active projector nonlinear gamma calibration,” Appl. Opt. 54(13), 3834–3841 (2015). [CrossRef]

17. Y. Xu, L. Ekstrand, J. Dai, and S. Zhang, “Phase error compensation for three-dimensional shape measurement with projector defocusing,” Appl. Opt. 50(17), 2572–2581 (2011). [CrossRef] [PubMed]

18. P. Zhou, X. Liu, Y. He, and T. Zhu, “Phase error analysis and compensation considering ambient light for phase measuring profilometry,” Opt. Lasers Eng. 55(7), 99–104 (2014). [CrossRef]

19. Z. Li and Y. Li, “Gamma-distorted fringe image modeling and accurate gamma correction for fast phase measuring profilometry,” Opt. Lett. 36(2), 154–156 (2011). [CrossRef] [PubMed]

20. K. Liu, Y. Wang, D. L. Lau, Q. Hao, and L. G. Hassebrook, “Gamma model and its analysis for phase measuring profilometry,” J. Opt. Soc. Am. A 27(3), 553–562 (2010). [CrossRef] [PubMed]

21. Z. Zhang and E. Zhang, “Fluctuation elimination of fringe pattern by using empirical mode decomposition,” Proc. SPIE 9046, 90460D (2013). [CrossRef]

22. G. Lu, S. Wu, N. Palmer, and H. Liu, “Application of phase shift optical triangulation to precision gear gauging,” Proc. SPIE 3520, 52–63 (1998). [CrossRef]

23. Y. Lu, R. Zhang, and H. Guo, “Correction of illumination fluctuations in phase-shifting technique by use of fringe histograms,” Appl. Opt. 55(1), 184–197 (2016). [CrossRef] [PubMed]

24. X. Xu, L. Cai, Y. Wang, X. Meng, X. Cheng, H. Zhang, G. Dong, and X. Shen, “Correction of wavefront reconstruction errors caused by light source intensity instability in phase-shifting interferometry,” J. Opt. A 10(8), 085008 (2008). [CrossRef]

25. V. Srinivasan, H. C. Liu, and M. Halioua, “Automated phase-measuring profilometry: a phase mapping approach,” Appl. Opt. 24(2), 185 (1985). [CrossRef] [PubMed]

26. Y. Fu and Q. Luo, “Fringe projection profilometry based on a novel phase shift method,” Opt. Express 19(22), 21739–21747 (2011). [CrossRef] [PubMed]

27. X. Su, S. Song, Y. Cao, and L. Xiang, “Phase-height mapping and coordinate calibration simultaneously in phase-measuring profilometry,” Opt. Eng. 43(3), 708–712 (2004). [CrossRef]

28. S. Liu, W. Feng, Q. Zhang, and Y. Liu, “Three-dimensional shape measurement of small object based on tri-frequency heterodyne method,” Proc. SPIE 9623, 96231C (2015). [CrossRef]

Instability of projection light source and real-time phase error correction method for phase-shifting profilometry

Abstract

1. Introduction

2. Principle for error induced by the instability of the projection light source

2.1 Instability of projection light source during experimentation

2.2 Mathematical model for time-dependent phase error induced by IPLS

3. Correction of IPLS induced time-dependent error

3.1 Phase error correction methods

3.2 Experiments

4. Discussion

5. Conclusion

Funding

Acknowledgments

References and links

Cited By

Figures (10)

Equations (13)

Optics Express