Flexible phase error compensation based on Hilbert transform in phase shifting profilometry

Zewei Cai; Xiaoli Liu; Hao Jiang; Dong He; Xiang Peng; Shujun Huang; Zonghua Zhang

doi:10.1364/OE.23.025171

1. Introduction

Phase shifting profilometry (PSP) has been developed rapidly for three dimensional (3D) shape measurement with the advantage of high-speed, high-resolution and high-accuracy [1–3 ]. In PSP system, some error sources, such as sensor noise, nonlinear response and quantization error, are introduced by the commercially available digital devices [4–6 ]. The nonlinear response named gamma effect, as a main error source, deviates a standard fringe pattern from ideal sinusoidal signal and brings considerable phase error into PSP. Recently various phase error compensation methods have been proposed [7–23 ]. However, some auxiliary conditions are required to achieve the goal of phase error compensation. For example, one sort of the compensation methods depends on a calibration procedure to quantify the nonlinearity of system response by using response curve fitting [7–9 ], phase benchmark building [10, 11 ], gamma model derivation [12–17 ], etc. Although the phase error caused by the nonlinear response can be compensated effectively with these methods, the calibration procedure is time-consuming and laborious. Moreover, because the operations of the calibration and measurement are sequential, this can lead to the disturbances in invalid calibration results and result in the growth of the phase error. In addition, other kind of non-calibration error compensation methods relied on defocusing technology [18–23 ]. The defocusing can actually work as a low-pass filter to eliminate high-frequency harmonics. However, it is a difficult task to control the degree of defocusing precisely, as mentioned in [24, 25 ].

To address the aforementioned issues, we try to investigate a flexible and robust compensation method against environment instabilities and independence on additional auxiliary conditions. The Hilbert transform (HT) has been introduced to extract the phase information from single-frame fringe pattern [26–34 ]. The main characteristic of the HT is that it will induce a phase shift of $π / 2$ to a signal with its amplitude remained but its DC component removed [27]. Considering the periodical distribution of the phase error in spatial domain [13], it can be predicted that the phase error distribution in HT domain should have the similar features. Furthermore, because of the phase shift induced by the HT, it is likely to make the phase errors have opposite distributional tendency in spatial domain and HT domains, respectively.

In this paper, the phase error models in spatial domain and HT domain are deduced explicitly, which are suitable to any step of PSP. Then we analyzed their characteristics and verify the prediction that the phase errors in the two domains are both periodic distributions with identical amplitude and opposite direction. Therefore, the phase error can be compensated flexibly and simply by averaging the phases in the two domains. Experimental results demonstrated that it is suitable for flexible, high-robust and automatic phase error compensation without requiring any auxiliary condition or complicated computation.

2. Principle

2.1. Phase error model in spatial domain

In ideal case (i.e. linear intensity response), the intensity of the nth captured image in N-step PSP can be represented as

I_{n} (x, y) = A (x, y) + B (x, y) \cos [ϕ (x, y) + δ_{n}], n = 1, 2, \dots, N

where

A

and

B

denote the background and modulation intensity respectively,

ϕ

represents the phase information modulated by the object depth, and

δ_{n} = 2 π (n - 1) / N

is the phase-shifting amount. For simplicity, we omit notation

(x, y)

hereafter.

In practice, the intrinsic nonlinear intensity response of a projector-camera setup introduces the high-order harmonics into the fundamental harmonic of fringe patterns. According to power-law response [35], the gamma-distorted intensity can be represented as [13]

I_{n}^{C} = {(α I_{n})}^{γ} = B_{0} + \sum_{k = 1}^{\infty} {B_{k} \cos [k (ϕ + δ_{n})]} = B_{0} + \sum_{k = 1}^{\infty} [B_{k} \cos (k ϕ_{n})]

where

α

denotes the surface reflectivity of objects,

B_{0}

and

B_{k}

are the DC component and the magnitude of the k-order harmonic component respectively,

ϕ_{n}

represents the modulated phase coupled with the phase shift, and

γ

is the gamma factor.

Based on the least square algorithm (LSA), the modulated phase can be worked out as

ϕ = arc \tan [\frac{- \sum_{n = 1}^{N} (I_{n} \sin δ_{n})}{\sum_{n = 1}^{N} (I_{n} \cos δ_{n})}]

Due to the power-law response, the gamma-distorted LSA-based phase is

\begin{matrix} ϕ^{C} = arc \tan [\frac{- \sum_{n = 1}^{N} (I_{n}^{C} \sin δ_{n})}{\sum_{n = 1}^{N} (I_{n}^{C} \cos δ_{n})}] = arc \tan {\frac{- B_{0} \sum_{n = 1}^{N} \sin δ_{n} - \sum_{n = 1}^{N} \sum_{k = 1}^{\infty} [B_{k} \cos (k ϕ_{n}) \sin δ_{n}]}{B_{0} \sum_{n = 1}^{N} \cos δ_{n} + \sum_{n = 1}^{N} \sum_{k = 1}^{\infty} [B_{k} \cos (k ϕ_{n}) \cos δ_{n}]}} \\ = arc \tan {\frac{- \sum_{n = 1}^{N} \sum_{k = 1}^{\infty} [B_{k} \cos (k ϕ_{n}) \sin δ_{n}]}{\sum_{n = 1}^{N} \sum_{k = 1}^{\infty} [B_{k} \cos (k ϕ_{n}) \cos δ_{n}]}} \end{matrix}

Considering

δ_{n} = 2 π (n - 1) / N

, it can be proved easily that

\sum_{n = 1}^{N} \sin δ_{n} = 0

and

\sum_{n = 1}^{N} \cos δ_{n} = 0

, so that the term

B_{0}

in Eq. (4) is eliminated. That shows the terms with respect to DC component can be eliminated with the LSA-based PSP.

We directly derive a phase error model of PSP system in a simple way as follow:

\begin{matrix} Δ ϕ = ϕ^{C} - ϕ \\ = arc \tan {\frac{- \cos ϕ \sum_{n = 1}^{N} \sum_{k = 1}^{\infty} [B_{k} \cos (k ϕ_{n}) \sin δ_{n}] - \sin ϕ \sum_{n = 1}^{N} \sum_{k = 1}^{\infty} [B_{k} \cos (k ϕ_{n}) \cos δ_{n}]}{\cos ϕ \sum_{n = 1}^{N} \sum_{k = 1}^{\infty} [B_{k} \cos (k ϕ_{n}) \cos δ_{n}] - \sin ϕ \sum_{n = 1}^{N} \sum_{k = 1}^{\infty} [B_{k} \cos (k ϕ_{n}) \sin δ_{n}]}} \\ = arc \tan {\frac{- \sum_{n = 1}^{N} \sum_{k = 1}^{\infty} [B_{k} \cos (k ϕ_{n}) \sin ϕ_{n}]}{\sum_{n = 1}^{N} \sum_{k = 1}^{\infty} [B_{k} \cos (k ϕ_{n}) \cos ϕ_{n}]}} \\ = arc \tan {\frac{\sum_{n = 1}^{N} \sum_{k = 2}^{\infty} [(B_{k + 1} - B_{k - 1}) \sin (k ϕ_{n})]}{N B_{1} + \sum_{n = 1}^{N} \sum_{k = 2}^{\infty} [(B_{k + 1} + B_{k - 1}) \cos (k ϕ_{n})]}} \end{matrix}

Considering

\begin{array}{l} \sum_{n = 1}^{N} [\sin (k ϕ_{n})] = {\begin{matrix} 0, k \neq m N \\ N \sin (m N ϕ), k = m N \end{matrix}, m \in Z^{+} \\ \sum_{n = 1}^{N} [\cos (k ϕ_{n})] = {\begin{matrix} 0, k \neq m N \\ N \sin (m N ϕ), k = m N \end{matrix}, m \in Z^{+} \end{array}

Equation (5) can be simplified as

Δ ϕ = arc \tan {\frac{\sum_{m = 1}^{\infty} [(G_{m N + 1} - G_{m N - 1}) \sin (m N ϕ)]}{1 + \sum_{m = 1}^{\infty} [(G_{m N + 1} + G_{m N - 1}) \cos (m N ϕ)]}}

where

G_{s} = B_{s} / B_{1}

is related to the gamma factor and can be expressed as [13]

G_{s} = \prod_{i = 2}^{s} \frac{γ - i + 1}{γ + i}

The value of

G_{s}

observably decreases with increasing harmonic order

s

. Actually, it is enough to consider N-order harmonic. Furthermore,

G_{N + 1}

can be neglected because of its little contribution to the gamma correction. Therefore, Eq. (7) can be further simplified as

Δ ϕ = arc \tan [\frac{- G_{N - 1} \sin (N ϕ)}{1 + G_{N - 1} \cos (N ϕ)}]

Equation (9) represents the phase error model based on the power-law response of PSP system in spatial domain. It indicates that the phase error caused by the gamma effect is a periodical distribution with respect to phase

ϕ

. We call it the universal model with the advantage of being suitable for arbitrary phase-shifting step. In order to develop a flexible phase error compensation method, another phase error model in HT domain is presented in the following subsection.

2.2. Phase error model in HT domain

The HT of the nth captured phase-shifting image can be represented as

I_{n}^{H} = H [I_{n}] = - B \sin (ϕ + δ_{n})

where

Η [•]

denotes an operator of HT. In the transform procedure, the DC component, i.e.

B_{0}

, may not be filtered out completely and remained in the transformed image. Fortunately its effect will be eliminated in the LSA-based PSP, as mentioned in spatial domain. In addition, it should be noticed that HT will bring error into the transformed images when the fringe area on object is less than one period. However it is an extreme condition. In practice, the dense fringe projection is generally used for achieving better accuracy.

Due to the gamma effect, there are also high-order harmonics existing in the transformed intensity, which can be represented as

I_{n}^{H C} = - \sum_{k = 1}^{\infty} [B_{k} \sin (k ϕ_{n})]

To make equal to $ϕ$ , the LSA-based phase in HT domain can be expressed by

ϕ^{H} = arc \tan [\frac{\sum_{n = 1}^{N} (I_{n}^{H} \cos δ_{n})}{\sum_{n = 1}^{N} (I_{n}^{H} \sin δ_{n})}]

The gamma-distorted phase in HT domain is

ϕ^{H C} = arc \tan [\frac{\sum_{n = 1}^{N} (I_{n}^{H C} \cos δ_{n})}{\sum_{n = 1}^{N} (I_{n}^{H C} \sin δ_{n})}] = arc \tan {\frac{\sum_{n = 1}^{N} \sum_{k = 1}^{\infty} [B_{k} \sin (k ϕ_{n}) \cos δ_{n}]}{\sum_{n = 1}^{N} \sum_{k = 1}^{\infty} [B_{k} \sin (k ϕ_{n}) \sin δ_{n}]}}

Similarly, we deduce the phase error model of PSP system in HT domain as

Δ ϕ^{H} = ϕ^{H C} - ϕ = arc \tan {\frac{\sum_{m = 1}^{\infty} [(G_{m N + 1} + G_{m N - 1}) \sin (m N ϕ)]}{1 + \sum_{m = 1}^{\infty} [(G_{m N + 1} - G_{m N - 1}) \cos (m N ϕ)]}}

The same simplifications as Eq. (7) are operated to Eq. (14), so the phase error model can be represented as

Δ ϕ^{H} = arc \tan [\frac{G_{N - 1} \sin (N ϕ)}{1 - G_{N - 1} \cos (N ϕ)}]

Equation (15) represents the phase error distribution in HT domain, which is also a periodical distribution related to the same factors as that in spatial domain.

3. Phase error compensation method

3.1. Characteristics of phase error distribution

Based on above phase error models, the characteristics of phase error distribution in the two domains can be discussed and analyzed. Equations (9) and (15) reveal that the phase errors in spatial domain and HT domain are trigonometric functions with respect to phase $ϕ$ , whose periods are both $T = 2 π / N$ . Let $\partial Δ ϕ / \partial ϕ = 0$ in Eq. (9) and $\partial Δ ϕ^{H} / \partial ϕ = 0$ in Eq. (15), the amplitude of the phase error distributions, namely the maximum phase errors, in the two domains can be obtained respectively by Eqs. (16) and (17)

A_{Δ ϕ} = {| Δ ϕ |}_{\max} = arc \sin (| G_{N - 1} |)

A_{Δ ϕ}^{H} = {| Δ ϕ^{H} |}_{\max} = arc \sin (| G_{N - 1} |)

Moreover, we can derive the relation between the Eqs. (9) and (15) as follow:

{Δ ϕ^{H} |}_{ϕ + \frac{T}{2}} = arc \tan {\frac{G_{N - 1} \sin [N (ϕ + \frac{π}{N})]}{1 - G_{N - 1} \cos [N (ϕ + \frac{π}{N})]}} = arc \tan [\frac{- G_{N - 1} \sin (N ϕ)}{1 + G_{N - 1} \cos (N ϕ)}] = {Δ ϕ |}_{ϕ}

Then, we can summarize the characteristics of the periodic phase error distributions in the two domains:

(1) The same period of $T = 2 π / N$ ;
(2) The same amplitude of $A = arc \sin (| G_{N - 1} |)$ , as shown in Eqs. (16) and (17) ;
(3) The distributions are half period shift, as shown in Eq. (18), leading to the opposite distributional tendency.

To illustrate these characteristics, we created some phase error curves in the two domains through Eqs. (9) and (15) with the parameters of $G_{N - 1} = 0.4$ and $N = 3$ , as demonstrated in Fig. 1(a) . It is clearly shown that the three stated characteristics are correct, and the values of phase error in the two domains are approximately identical in magnitude and opposite in sign.

Fig. 1 Characteristics of phase error distribution: (a) the phase error with respect to phase, (b) the maximum phase error with respect to phase-shifting step.

Download Full Size | PDF

3.2. Compensation for phase errors

Let $ϕ^{M} = (ϕ^{C} + ϕ^{H C}) / 2$ denote the average phase between the two domains. Combining Eqs. (9) and (15) , the phase error of $ϕ^{M}$ can be obtained as

Δ ϕ^{M} = ϕ^{M} - ϕ = \frac{1}{2} (Δ ϕ + Δ ϕ^{H}) = \frac{1}{2} arc \tan [\frac{G_{N - 1}^{2} \sin (2 N ϕ)}{1 - G_{N - 1}^{2} \cos (2 N ϕ)}]

It is still a periodical distribution with two-fold frequency in spatial domain, as demonstrated in Fig. 1(a). Let

\partial Δ ϕ^{M} / \partial ϕ = 0

in Eq. (19), the maximum phase error of the average phase is

{| Δ ϕ^{M} |}_{\max} = \frac{1}{2} arc \sin (G_{N - 1}^{2})

Compared Eq. (20) with Eq. (16) and (17) , the maximum phase error of the average phase is reduced, because

| G_{N - 1} | < 1

.

In order to quantify the reduction of the maximum phase error, we use the Taylor series expansion of arcsine function considered only the first term, i.e. $\arcsin (x) = x + 1 / 6 x^{3} + \circ (x^{5}) \approx x, | x | < 1$ . The ratio of the maximum phase errors is approximated as

r_{Δ ϕ} = \frac{{| Δ ϕ^{M} |}_{\max}}{{| Δ ϕ |}_{\max}} \approx \frac{\frac{1}{2} G_{N - 1}^{2}}{| G_{N - 1} |} = \frac{| G_{N - 1} |}{2}

It reveals that after average operation the maximum phase error can be reduced to $| G_{N - 1} | / 2$ times of the original one. According to Eqs. (16) and (20) , the characteristics of the maximum phase error with respect to various phase-shifting steps and gamma factors are showed in Fig. 1(b). Besides, the relevant data statistics are listed in Table 1 . It is clear that in 3-step PSP the phase error associated with nonlinear response can be reduced by about 80% after average operation, while in four- or more step PSP that can be reduced by more than 95% in the simulation calculation.

Table 1. Maximum phase error (rad) and ratio of error reduction (%)

View Table | View all tables in this article

Therefore, we propose a phase error compensation method summarized as follow:

Step 1, transform the phase-shifting fringe images in spatial domain to HT domain
Step 2, compute two phases in spatial domain and HT domain using the LSA of Eqs. (3) and (12) , respectively.
Step 3, average the phases in the two domains to generate the corrected phase.

4. Experiments and analysis

We developed a PSP system combining a DLP projector (DELL, M110) and a CMOS camera (DH, MER-130-30UM) with a 16-mm imaging lens (PENTAX, C1614-M) to demonstrate the validity of the proposed method. A white board was taken to be a target. In the experiments, we performed 1D HT algorithm row by row to the captured images in vertical fringe projection. In our 1D HT algorithm, the fast Fourier transform (FFT) of the input discrete signal was calculated firstly. Then those FFT coefficients of negative frequencies were replaced with zeroes, and the inverse FFT was implemented to achieve the result finally. More details can be referred to literature [36]. As the implementation of HT depends on the FFT, some defects such as signals aliasing, truncation effect, spectrum leakage, and so on, may arise in this HT algorithm. Additionally, in the processing of the discrete HT, the sampling of signum function may lead to an approximate result with weak edge effect. The transformed fringe images were used to compute phase in HT domain.

Figure 2 shows one cross section of the phase error distributions with and without compensation. As mentioned before, the phase error distributions in the two domains are identical in amplitude and opposite in direction. In this sense, the phase error is effectively compensated through the average operation. As shown in Fig. 2(a), the experimental results of three-step PSP greatly match with the above features. However, in Fig. 2(b), there is a little mismatch about the phase error distribution after compensation in four-step PSP comparing with the simulation calculation. In our opinion, this mismatch may come from the impacts of other error sources, such as sensor noise, quantization error, ambient light disturbance, because the effect of nonlinear response will be decreased rapidly with the increase of the phase-shifting step, on the other hand, the impact of other error sources will become more significant.

Fig. 2 One cross section of the phase error distribution with and without compensation: (a) in three-step PSP, and (b) four-step PSP.

Download Full Size | PDF

Table 2 lists the maximum (MAX) and root-mean-square (RMS) values of the phase error with and without compensation. It clearly shows the effectiveness of the proposed compensation method.

Table 2. MAX and RMS of the phase error (rad)

View Table | View all tables in this article

Another experiment is a 3D digital shape reconstruction of a plaster statue with free-form surface. Figure 3 shows the captured images with uniform and fringe projection illumination. Figure 4 and 5 show the 3D reconstruction results by using three-step and four-step PSP, respectively. In the reconstruction procedure, the 3D coordinates of points on measured object are computed firstly with the phase and calibration information. Then combining the neighboring relations of those points, we can generate the 3D model with triangle meshes. It can be seen that the periodical ripple effect distinctly appears on the reconstructed 3D digital surface because of the nonlinear response. With the phase error compensation, the ripple effect is improved significantly.

Fig. 3 The captured images: (a) in uniform illumination, and (b) fringe projection illumination.

Download Full Size | PDF

Fig. 4 3D digital reconstruction results by using three-step PSP: (a) without compensation in spatial domain, (b) without compensation in HT domain, (c) with compensation by the proposed method, (d)-(f) the corresponding 3D digital surface details in (a)-(c), and (g) a local cross section of 3D digital surface, as labeled in Fig. 3(a).

Download Full Size | PDF

Fig. 5 3D digital reconstruction results by using four-step PSP: (a) without compensation in spatial domain, (b) without compensation in HT domain, (c) with compensation by the proposed method, (d)-(f) the corresponding 3D digital surface details in (a)-(c), and (g) a local cross section of 3D digital surface, as labeled in Fig. 3(a).

Download Full Size | PDF

In order to evaluate the time efficiency of our method on the processing of HT and phase retrieval, we record the time cost of the experiments. The testing environments are a CPU of i3-4130, a RAM of 8G, and software of MATLAB R2015a. The time costs of performing HT in 3- and 4-step PSP are 1256ms and 1633ms, while that of phase retrieval are 539ms and 596ms. It is obvious that the proposed method just requires a little extra time to perform the phase error compensation automatically and flexibly, without needing for additional information or operation.

5. Conclusions

This paper derived the phase error models respectively in spatial domain and HT domain and then analyzed and compared their characteristics of phase error distribution. It was found that the phase error distributions in the two domains are identical in amplitude and opposite in direction. According to this phenomenon, a simple and flexible phase error compensation method was proposed by only using the average phase between the two domains. Theoretical analysis and experimental results indicated that the phase error can be reduced by about 80% in three-step PSP, and more than 95% in four or more step PSP.

Acknowledgments

We gratefully acknowledge the financial support from the Natural Science Foundation of China (NSFC) under Grant No. 61201355, 61171048, 61377017, 61311130138, 61405122 and the Sino-German Center for Research Promotion (SGCRP) under Grant No. GZ 760. The grants from Scientific and Technological Project of the Shenzhen government(JCYJ20140828163633999), Key Basic Research Project of Applied Basic Research Programs Supported by Hebei Province (15961701D), and Research Project for High-level Talents in Hebei University (GCC2014049) are also acknowledged.

References and links

1. C. Quan, X. Y. He, C. F. Wang, C. J. Tay, and H. M. Shang, “Shape measurement of small objects using LCD fringe projection with phase shifting,” Opt. Commun. 189(1–3), 21–29 (2001). [CrossRef]

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

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

4. C. Rathjen, “Statistical properties of phase-shift algorithms,” J. Opt. Soc. Am. A 12(9), 1997–2008 (1995). [CrossRef]

5. G. H. Notni and G. Notni, “Digital fringe projection in 3D shape measurement: an error analysis,” Proc. SPIE 5144, 372–380 (2003). [CrossRef]

6. Z. Wang, D. A. Nguyen, and J. C. Barnes, “Some practical considerations in fringe projection profilometry,” Opt. Lasers Eng. 48(2), 218–225 (2010). [CrossRef]

7. C. R. Coggrave and J. M. Huntley, “High-speed surface profilometer based on a spatial light modulator and pipeline image processor,” Opt. Eng. 38(9), 1573–1581 (1999). [CrossRef]

8. S. Kakunai, T. Sakamoto, and K. Iwata, “Profile measurement taken with liquid-crystal gratings,” Appl. Opt. 38(13), 2824–2828 (1999). [CrossRef] [PubMed]

9. P. S. Huang, C. Zhang, and F. Chiang, “High-speed 3-D shape measurement based on digital fringe projection,” Opt. Eng. 42(1), 163–168 (2003). [CrossRef]

10. S. Zhang and S. T. Yau, “Generic nonsinusoidal phase error correction for three-dimensional shape measurement using a digital video projector,” Appl. Opt. 46(1), 36–43 (2007). [CrossRef] [PubMed]

11. B. Pan, Q. Kemao, L. Huang, and A. Asundi, “Phase error analysis and compensation for nonsinusoidal waveforms in phase-shifting digital fringe projection profilometry,” Opt. Lett. 34(4), 416–418 (2009). [CrossRef] [PubMed]

12. T. Hoang, B. Pan, D. Nguyen, and Z. Wang, “Generic gamma correction for accuracy enhancement in fringe-projection profilometry,” Opt. Lett. 35(12), 1992–1994 (2010). [CrossRef] [PubMed]

13. 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]

14. 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]

15. X. Zhang, L. Zhu, Y. Li, and D. Tu, “Generic nonsinusoidal fringe model and gamma calibration in phase measuring profilometry,” J. Opt. Soc. Am. A 29(6), 1047–1058 (2012). [CrossRef] [PubMed]

16. S. Ma, C. Quan, R. Zhu, L. Chen, B. Li, and C. J. Tay, “A fast and accurate gamma correction based on Fourier spectrum analysis for digital fringe projection profilometry,” Opt. Commun. 285(5), 533–538 (2012). [CrossRef]

17. H. Cui, Z. Zhao, Y. Wu, N. Dai, X. Cheng, and L. Zhang, “Digital fringe image gamma modeling and new algorithm for phase error compensation,” Optik (Stuttg.) 125(24), 7175–7181 (2014). [CrossRef]

18. X.-Y. Su, W.-S. Zhou, G. von Bally, and D. Vukicevic, “Automated phase-measuring profilometry using defocused projection of a Ronchi grating,” Opt. Commun. 94(6), 561–573 (1992). [CrossRef]

19. M. J. Baker, J. Xi, and J. F. Chicharo, “Elimination of gamma non-linear luminance effects for digital video projection phase measuring profilometers”, in 4th IEEE International Symposium on Electronic Design, Test and Applications (IEEE, 2008), pp. 496–501. [CrossRef]

20. S. Lei and S. Zhang, “Flexible 3-D shape measurement using projector defocusing,” Opt. Lett. 34(20), 3080–3082 (2009). [CrossRef] [PubMed]

21. Y. Wang and S. Zhang, “Optimal pulse width modulation for sinusoidal fringe generation with projector defocusing,” Opt. Lett. 35(24), 4121–4123 (2010). [CrossRef] [PubMed]

22. 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]

23. D. Zheng and F. Da, “Absolute phase retrieval for defocused fringe projection three-dimensional measurement,” Opt. Commun. 312, 302–311 (2014). [CrossRef]

24. S. Zhang, Y. Gong, Y. Wang, J. Laughner, and I. G. Efimov, “Some recent advance on high-speed, high-resolution 3-D shape measurement using projector defocusing”, in 2010 International Symposium on Optomechatronic Technologies (IEEE, 2010), pp. 1–6 [CrossRef]

25. B. Li, Y. Wang, J. Dai, W. Lohry, and S. Zhang, “Some recent advances on superfast 3D shape measurement with digital binary defocusing techniques,” Opt. Lasers Eng. 54, 236–246 (2014). [CrossRef]

26. D. A. Zweig and R. E. Hufnagel, “Hilbert transform algorithm for fringe-pattern analysis,” Proc. SPIE 1333, 295–302 (1990). [CrossRef]

27. M. A. Sutton, W. Zhao, S. R. McNeill, H. W. Schreier, and Y. J. Chao, “Development and assessment of a single-image fringe projection method for dynamic applications,” Exp. Mech. 41(3), 205–217 (2001). [CrossRef]

28. K. G. Larkin, D. J. Bone, and M. A. Oldfield, “Natural demodulation of two-dimensional fringe patterns. I. General background of the spiral phase quadrature transform,” J. Opt. Soc. Am. A 18(8), 1862–1870 (2001). [CrossRef] [PubMed]

29. K. G. Larkin, “Natural demodulation of two-dimensional fringe patterns. II. Stationary phase analysis of the spiral phase quadrature transform,” J. Opt. Soc. Am. A 18(8), 1871–1881 (2001). [CrossRef] [PubMed]

30. V. Madjarova, H. Kadono, and S. Toyooka, “Dynamic electronic speckle pattern interferometry (DESPI) phase analyses with temporal Hilbert transform,” Opt. Express 11(6), 617–623 (2003). [CrossRef] [PubMed]

31. V. D. Madjarova, H. Kadono, and S. Toyooka, “Use of dynamic electronic speckle pattern interferometry with the Hilbert transform method to investigate thermal expansion of a joint material,” Appl. Opt. 45(29), 7590–7596 (2006). [CrossRef] [PubMed]

32. U. P. Kumar, N. K. Mohan, and M. P. Kothiyal, “Time average vibration fringe analysis using Hilbert transformation,” Appl. Opt. 49(30), 5777–5786 (2010). [CrossRef] [PubMed]

33. M. Trusiak, K. Patorski, and K. Pokorski, “Hilbert-Huang processing for single-exposure two-dimensional grating interferometry,” Opt. Express 21(23), 28359–28379 (2013). [CrossRef] [PubMed]

34. M. Trusiak, K. Patorski, and M. Wielgus, “Hilbert-Huang processing and analysis of complex fringe patterns,” Proc. SPIE 9203, 92030K (2014). [CrossRef]

35. C. A. Poynton, “‘Gamma’ and its disguises: the nonlinear mappings of intensity in perception, CRTs, film, and video,” SMPTE Journal 102(12), 1099–1108 (1993). [CrossRef]

36. S. L. Marple, “Computing the discrete-time ‘analytic’ signal via FFT,” IEEE Trans. Signal Process. 47(9), 2600–2603 (1999). [CrossRef]

gamma	3-step				4-step				5-step
gamma	N	Y	R		N	Y	R		N	Y	R
2.2 2.5 2.8 3.1	0.2898 0.3398 0.3844 0.4244	0.0409 0.0557 0.0705 0.0852	85.90 83.62 81.65 79.93		0.0110 0.0303 0.0517 0.0743	6.04e-5 4.59e-4 0.0013 0.0028	99.45 98.49 97.41 96.29		0.0014 0.0023 0.0015 0.0010	1.00e-6 2.72e-6 1.16e-6 5.47e-7	99.93 99.88 99.92 99.95

	Without compensation				With compensation
	spatial domain		HT domain		With compensation
N	MAX	RMS	MAX	RMS	MAX	RMS
3	0.3700	0.2354	0.3503	0.2325	0.0745	0.0396
4	0.0436	0.0197	0.0459	0.0217	0.0165	0.0063

gamma	3-step				4-step				5-step
gamma	N	Y	R		N	Y	R		N	Y	R
2.2 2.5 2.8 3.1	0.2898 0.3398 0.3844 0.4244	0.0409 0.0557 0.0705 0.0852	85.90 83.62 81.65 79.93		0.0110 0.0303 0.0517 0.0743	6.04e-5 4.59e-4 0.0013 0.0028	99.45 98.49 97.41 96.29		0.0014 0.0023 0.0015 0.0010	1.00e-6 2.72e-6 1.16e-6 5.47e-7	99.93 99.88 99.92 99.95

	Without compensation				With compensation
	spatial domain		HT domain		With compensation
N	MAX	RMS	MAX	RMS	MAX	RMS
3	0.3700	0.2354	0.3503	0.2325	0.0745	0.0396
4	0.0436	0.0197	0.0459	0.0217	0.0165	0.0063

Flexible phase error compensation based on Hilbert transform in phase shifting profilometry

Abstract

1. Introduction

2. Principle

2.1. Phase error model in spatial domain

2.2. Phase error model in HT domain

3. Phase error compensation method

3.1. Characteristics of phase error distribution

3.2. Compensation for phase errors

4. Experiments and analysis

5. Conclusions

Acknowledgments

References and links

Cited By

Figures (5)

Tables (2)

Equations (21)

Optics Express