Spatial quasi-phase-shifting technique for single-frame dynamic fringe analysis

Zibang Zhang; Jingang Zhong

doi:10.1364/OE.22.002695

1. Introduction

Phase demodulation plays an important role in fringe analysis. A major goal of the fringe analysis is to demodulate the phase distribution from the fringe intensity distribution, since the fringe phase is modulated by the physical quantity to be measured. There exist several methods for phase demodulation such as Fourier transform (FT) method [1], Windowed Fourier transform method [2], wavelet transform (WT) method [3], phase-shifting (PS) technique [4], etc. The FT method, one of the most commonly used methods, requires one fringe pattern for demodulation. It is a global operation in frequency domain and is, therefore, not localized in spatial domain. The absence of spatial localization might lead to spectral overlapping in the frequency domain. In that case, the first-order component can be hardly accurately extracted. To improve the spatial localization, the windowed FT method and the WT method were proposed. Both of them have satisfactory spatial localization which is at a several-fringe-period level. For the windowed FT method, its spatial localization is determined by the width of a fixed Gaussian window. The automatic selection of window, including the first-order component extraction window for the FT method and the Gaussian window for the windowed FT method, remains a problem and it is essential in dynamic applications. The WT method is adaptive due to the usage of scale factor. The spatial localization of the WT method is at a several-period level and is adjusted to carrier frequency. It suffers a poor spatial localization when the carrier frequency is relatively low [5]. The PS technique is well known for its single-point-level spatial localization by recoding multiple temporally adjacent fringe patterns. To adapt the PS technique to dynamic applications, one kind of approaches is to implement phase shifting within in a very short time, which is called superfast phase-shifting method. For instance, S. Zhang with his group has reported an implement of kilo-Hertz phase-shifting by using binary defocusing technique [6]. And another kind of approaches is to utilize the intra-frame phase increment, which is called spatial carrier phase-shifting method. M. Kujawinska and J. Wojciak proposed to introduce a large amount of tilt into an interferogram, making the phase difference between successive pixels equals $π / 2$ [7]. P. Chan, et al. proposed to subdivide a linearized fringe pattern into three component patterns and calculate the phase with the three-step phase-shifting algorithm [8]. Y. Du, et al. presented a non-iterative spatial phase-shifting algorithm by using principle component analysis [9]. Y. Awatsuji proposed a parallel quasi-phase-shifting (PQPS) technique which is capable of implementing approximate phase shifting in a 2 × 2-pixel area by using a specific phase-shifting device [10]. We propose a spatial quasi-phase-shifting (SQPS) technique which can be used in dynamic measurement without an extra phase-shifting device. Based on a new mathematical representation of fringe signal, it uses intra-frame phase shifts for demodulation.

2. Theory

2.1 Frequency-modulated representation of fringe signal

As the mathematical basis of the SQPS, a representation of fringe signal based on frequency modulation should be first introduced. Conventionally, a one-dimension (1-D) deformed fringe signal is represented in the phase-modulated form:

I (x) = a (x) + b (x) \cos [2 π f_{0} x + Δ ϕ (x)],

where

a (x)

is the direct current (DC) term resulted from the background illumination of the intensity distribution

I (x)

,

b (x)

is the modulated amplitude,

f_{0}

represents the carrier frequency and

Δ ϕ (x)

is the modulated phase. The fringe signal represented in the phase-modulated form reveals the physical content that the phase

Δ ϕ (x)

is modulated by the physical quantity to be measured. In the signal processing, a phase-modulated signal can be also expressed in the frequency-modulated form. Therefore, the fringe signal can be represented as a frequency-modulated signal:

I (x) = a (x) + b (x) \cos [2 π f_{L} (x) \cdot x + ϕ_{0, L} (x)],

where

f_{L} (x)

is the signal local frequency at point

x

and

ϕ_{0}_{,L} (x)

is the local initial phase corresponding to the local frequency. The term local frequency is also known as instantaneous frequency [11] in time-frequency analysis. The instantaneous frequency is used to quantify the time-frequency signature of mono-component non-stationary time-varying signal and the local frequency is for the space-varying signal. Comparing Eq. (1) with Eq. (2), we can obtain an equation as follows:

2 π f_{0} x + Δ ϕ (x) = 2 π f_{L} (x) \cdot x + ϕ_{0}_{,L} (x) .

The local frequency, as Fig. 1 shows, corresponds to the frequency of a sine wave (red solid line) which locally fits the signal (blue dotted line) under analysis. A fringe signal can be viewed as a spatial combination of slices of sine wave of different local frequencies. Considering the fringe in a small area, the local frequency changes so slightly that

I (x)

can be viewed as it shares a local frequency with spatially adjacent points. The phase shift between the adjacent points is an approximate constant. We call such a phase shift as the spatial quasi phase shift.

Fig. 1 Illustration of local frequency. Sine waves (solid lines) of different local frequencies fit fringe signal (dotted line) locally.

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2.2 Spatial quasi-phase-shifting technique

Assuming that the variations of the background term $a (x)$ and modulated amplitude $b (x)$ are negligible in a small area, the intensity distribution $I (x)$ can be simplified as:

I (x) = a + b \cos [2 π f_{L} (x) \cdot x + ϕ_{0}_{,L} (x)] .

By integrating the signal within three spatially adjacent intervals, we obtain three integrations with a fixed phase shift as follows:

{\begin{matrix} I_{1} = \int_{x - 3 k Δ x / 2}^{x - k Δ x / 2} I (x) d x, \\ = A + B \cos (2 π {\bar{f}}_{L} x + ϕ_{0, L} - 2 π {\bar{f}}_{L} k Δ x), \\ I_{2} = \int_{x - k Δ x / 2}^{x + k Δ x / 2} I (x) d x \\ = A + B \cos (2 π {\bar{f}}_{L} x + ϕ_{0, L}), \\ I_{3} = \int_{x + k Δ x / 2}^{x + 3 k Δ x / 2} I (x) d x \\ = A + B \cos (2 π {\bar{f}}_{L} x + ϕ_{0, L} + 2 π {\bar{f}}_{L} k Δ x), \end{matrix}

with

{\begin{cases} {\bar{f}}_{L} = \frac{1}{3 k Δ x} \cdot \int_{x - 3 k Δ x / 2}^{x + 3 k Δ x / 2} f_{L} (x) d x, \\ A = a k Δ x, \\ B = - 2 b \sin (π {\bar{f}}_{L} k Δ x), \end{cases}

where

Δ x

denotes the pixel size of a photosensor and

k

is the number of pixels in each integration interval (

k \in N^{+}

). According to Eq. (5), the phase shift of the three integrations is

2 π {\bar{f}}_{L} k Δ x

and it is dependent on the local frequency. As the local frequency can be obtained by relative techniques, such as short-time FT [12], WT [13], chirplet transform [14], Wigner-Ville distribution [15], etc., the phase term

(2 π {\bar{f}}_{L} x + ϕ_{0, L})

in Eq. (5) can be solved by adapting the typical three-step PS algorithm. And, by substituting the result into the right-hand side of Eq. (3), the desired

Δ ϕ (x)

is achieved:

Δ ϕ (x) = arc \tan [\frac{I_{1} - I_{3}}{2 I_{2} - I_{3} - I_{1}} \tan (k π {\bar{f}}_{L})] - 2 π f_{0} x .

Figure 2 illustrates the three-adjacent-neighbor demodulation of the SQPS.

Fig. 2 Illustration of SQPS: (a) k = 3 and (b) k = 1.

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One can make a compromise between the spatial localization and the anti-noise property by tuning the parameter k which determines the interval of integration. When $k = 1$ , the integration interval consists of only one point (see Fig. 2(b)). In that case, the SQPS implements the demodulation with three spatially adjacent points, providing the best spatial localization in principle. Considering the existence of noise, the variation of the fringe intensity $I (x)$ in a small neighbor might be drowned in the noise, which results in pronounced demodulation errors. Note that the noise here is referred in particular to the white noise, corresponding to thermal noise and shot noise, which is a main source of photosensors noise. Tuning the parameter k greater to enlarge the integration intervals is beneficial to eliminating the noise. The integration interval should be smaller than a fringe period, that is, one has to ensure $k \leq 1 / 3 Δ x {\bar{f}}_{L}$ . On one hand, a greater k results in larger integration intervals that are beneficial to evening out the noise. On the other hand, the greater k leads to poorer spatial localization, and consequently, lower accuracy. It is because the averaging effect of integration leads to the loss of signal details. Therefore, one has to make a compromise between the spatial localization and the anti-noise property.

Note that the phase can be simply achieved, in principle, by computing a numerical integral of detected local frequency [16], if the local frequency is accurately detected. But small errors would be accumulated due to the integral operation. Accurate local frequency detection is still a difficult problem in practice. So far most detecting methods can only achieve an approximate result. Therefore, it is meaningful of the proposed technique, combing a local frequency detection method with three-step phase-shifting algorithm, which is capable of achieving accurate phase with an approximately estimated local frequency.

3. Numerical simulation

We generated a 1-D 512-point fringe signal for the numerical simulation. The intensity distribution of the fringe can be expressed as follows:

I (x) = 127 + 127 \cdot \cos [2 π f_{0} x + Δ ϕ (x)],

where the coordinate

x = 1, 2, \dots, 512

, the carrier frequency

f_{0} = 0.049

cycle/point and the modulated phase

Δ ϕ (x)

is given by:

\begin{array}{l} Δ ϕ (x) = 3 {(1 - x / 512)}^{2} \exp [- {(x / 512)}^{2} - 1] \\ - 10 [5 x / 512 - {(x / 512)}^{3}] \exp [- {(x / 512)}^{2}] \\ - 1 / 3 \exp [- {(x / 512 + 1)}^{2}] . \end{array}

For the noiseless case, the resulting fringe is given in Fig. 3(a). Wavelet transform was used to detect the local frequency and the detection result is shown in Fig. 3(b). The phase was demodulated by the SQPS with k = 1, k = 3, and k = 7 respectively. The demodulation results and errors are given in Figs. 3(c) and 3(d). As both figures show, the SQPS achieves accurate demodulation in noiseless condition. The smaller k leads to the better demodulation, since it implies the better spatial localization, while good spatial localization is a prerequisite of good demodulation accuracy. Note that the errors on boundary are due to the boundary effect in the local frequency detection process, which can be efficiently dealt with the extrapolation of fringes.

Fig. 3 Simulation without noise: (a) generated 1-D fringe for demodulation, (b) detected local frequency by WT, (c) demodulated phase by SQPS with k = 1, k = 5, and k = 7 and (d) demodulation errors.

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For the noisy case, white noise was added into the fringe signal. In addition, the signal in the range $x \in [150, 180]$ was smeared to simulate a shadow. The resulting signal and the additional noise are given in Figs. 4(a.1) and 4(a.2), respectively. The local frequency of the signal was also detected by WT. With the detected local frequency as shown in Fig. 4(b), the SQPS with k = 1, k = 3, and k = 5 was employed. As the results shown in Figs. 4(c) and 4(d), the smaller k is, the more sensitive to noise the SQPS is. The errors due to the shadow are effectively isolated because of the spatial localization property of SQPS. Besides, for the result corresponding to k = 1, there exists a violent undulation in the range $x \in [350, 390]$ . It is actually a number of 2π jumps caused by the demodulation error near the boundary of the wrapped phase. To reduce the negative effect of noise in practice, one can employ a pre-process for noise reduction and increase the value of the parameter k to even out the noise.

Fig. 4 Simulation with noise: (a.1) generated 1-D fringe for demodulation, (a.2) additional white noise, (b) detected local frequency by WT, (c) demodulated phase by SQPS with k = 1, k = 3, and k = 5 and (d) demodulation errors.

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4. Experiment

We also tested the SQPS in fringe projection profilometry [17] in order to verify its performance in reality. The optical geometry of the experiment is given in Fig. 5, where $h$ represents object surface height from the reference plane, $l_{0}$ is the distance between the projector and the object, $d$ is the distance between the projector and the camera, ${E^{'}}_{p}$ and ${E^{'}}_{c}$ indicate the optical axis of the projector and the camera respectively. A digital projector was used to project a standard sinusoid fringe pattern onto an object surface. The resulting fringe pattern would be deformed due to the variation of the surface height. Substantially, the fringe phase was modulated by the height distribution. Therefore, the object surface can be reconstructed by demodulating the fringe phase.

Fig. 5 Optical geometry of fringe projection profilometry.

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4.1 Static measurement

We used a plaster model of a human head as the object with the experimental parameters $l_{0} = 1.05 m$ , $d = 0.22 m$ . The object had various height distributions. A $512 \times 512$ -pixel standard fringe pattern, with the carrier frequency 0.15 cycle/pixel, was projected onto the plaster model and the deformed fringe pattern was captured with a digital camera (Olympus C-770). The carrier frequency of the resultant fringe pattern on object surface was about 3.5 cycle/cm. A $1000 \times 1000$ -pixel area, where the object was located, was cropped from the captured image whose original size was 2288 (H) × 1712 (V) pixels. The cropped image is shown in Fig. 6.

Fig. 6 Captured deformed fringe pattern used in experiment (a) where red line indicates 516th column. (b) is intensity distribution of fringe at 516th column and (c) is its Fourier amplitude spectrum where dotted box represents the rectangle filter window used in FT and dash dot line indicates the cutoff frequency of low-pass filter.

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The 2-D deformed fringe phase was demodulated column by column. We chose the 516th column for comparison, where both rapid and slow variations exist. The intensity distribution and its Fourier amplitude spectrum are given in Figs. 6(b) and 6(c), respectively. As the spectrum shows, there exists overlapping resulting in the first-order component unable to be accurately extracted in the FT method. To filter the high frequency noise, corresponding to the generation-recombination noise, a low-pass filter was performed on the captured image as a pre-process. Note that the cutoff frequency of the low-pass filter can be coarsely determined as long as the first-order and the DC components are completely passed. For automatic selection of the cutoff frequency, in our case, it was chosen as 1.5 $f_{0}$ (see the green dash dot line in Fig. 6(c)), where $f_{0}$ indicates the carrier frequency of the captured fringe pattern. The phase demodulation was then implemented by using the SQPS (k = 1), the FT method, the WT method and the four-step PS technique, respectively. The rectangle window for extracting the first-order component in the FT method is given in Fig. 6(c). The local frequency of the fringe was detected by WT as shown in Fig. 7. With the phase unwrapped by the flood fill algorithm [18], the demodulation results of the 516th column are presented in Fig. 8.

Fig. 7 Local frequency of 516th column detected by WT.

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Fig. 8 Comparison of demodulation results at 516th column.

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The result by the FT method appears to be smooth, but there exists a global linear phase shift. Such a linear phase shift might result from the discretization error in carrier frequency removal and the error due to spectrum overlapping. Since the results of the other three methods overlap in Fig. 8, we choose two local areas, a slow-varying-frequency area and a rapid-varying-frequency area, for a further comparison. The one, shown in Fig. 9(a), is the area where the frequency varies slowly. The result of the SQPS appears to be as smoothed as that of the WT method. And the other, shown in Fig. 9(b), is the area where the frequency varies rapidly. The phase demodulated by the WT method is the most smoothed. However, the accuracy is poor, which represents that the top of the peak and the bottom of the valley cannot be reached. The results of the SQPS and the PS technique are of high accuracy. Though corrugated these two results appear to be, the envelopes are able to represent the profile of the object well. We also present the 3-D distributions of the full-frame demodulated phase in Fig. 10.

Fig. 9 Comparisons of demodulated results in (a) slow-varying-frequency area and (b) rapid-varying-frequency area.

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Fig. 10 3-D distributions of demodulated phases (a) by PS technique, (b) by SQPS (k = 1) technique, (c) by WT method, and (d) by FT method.

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Comparing Figs. 10(a)-10(c), the result of the PS method provides the most details while the result of the WT method does the least (see the eyes and the lips). The SQPS achieves the demodulation of the modest details and smoothness.

4.2 Dynamic measurement

A dynamic profilometry was also conducted to further verify the proposed technique. The same standard fringe pattern as that used in Section 4.1 was projected on a human face. The experimental parameters were $l_{0} = 1.34 m$ , $d = 0.20 m$ , according to the setup in Fig. 5. The carrier frequency of the resultant fringe pattern on object surface was about 3.25 cycle/cm. A digital camera (SONY XCD-SX910) was used to capture the fringe patterns deformed by human mouth in real time. The camera is with 1376 (H) × 1024 (V) pixels of size 4.65 µm (H) × 4.65 µm (V), operating at the rate of 15 frames per second. A 18-second long video was captured. Each frame of the video was cropped to be 800 (H) × 446 (V) pixels in size, where the human mouth was located. The SQPS (k = 3) was utilized each frame with a low pass filter as a pre-process. The cutoff frequency of the low pass filter was also 1.5 $f_{0}$ ( $f_{0}$ represents the carrier frequency of the captured fringe pattern). The demodulated phases were unwrapped by using the flood fill algorithm. The wavelet transform in local frequency detection process and the flood fill algorithm in the phase unwrapping process consumed around 5 seconds for each frame. As a result, the real-time acquired fringe patterns had to be processed frame by frame afterwards. The 3-D dynamic model of the mouth was reconstructed with the demodulated phases. Figure 11 shows the result of the 138th frame by using the proposed technique. Please see Media 1 for the whole demodulation and reconstruction result.

Fig. 11 Result of the dynamic profilometry: (a) 138th frame of captured video of deform fringe pattern, (b) 2-D distribution of demodulated phase, (c) 3-D presentation of demodulated phase, and (d) 2-D distribution of unwrapped phase.

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5. Conclusion

The spatial quasi-phase-shifting technique for fringe analysis is presented. A novel representation of fringe signal, based on frequency modulation, is also proposed. The proposed technique is a framework that can achieve phase demodulation with local frequency by using any local frequency detection methods. It operates in the spatial domain and is robust to noise. Requiring only one frame fringe pattern, the technique is capable of implementing dynamic measurements. Moreover, without any additional phase-shifting devices, it demodulates the fringe phase by using the intra-frame phase shifting even the local frequency is not so properly determined. The proposed technique has several-point level spatial localization, but a property of errors isolation as well. Both the simulation and experiment have shown the validity of the proposed technique. The technique can be applied in fringe projection profilometry, digital holography, Moiré interferometry, etc. It is expected that real-time measurement by the proposed technique could be achieved by using faster algorithms in the local frequency detection process and the phase unwrapping process. For instance, the short-time FT, which is less accurate than the wavelet transform, could be used for local frequency detection. But there exists a tradeoff between accuracy and computation. Another approach is employing technical optimization, such as parallel computation for processing multiple inter-frame columns (or rows) simultaneously, or utilizing the relevance of adjacent frame to reduce computation.

Acknowledgment

This work was supported by National Natural Science Foundation of China under the Grant No. 61077003 and No. 61307011. The authors appreciate Xiao Ma, Man Liang, and Qinqiu Fang for their assistance to the work.

References and links

1. M. Takeda, “Spatial-carrier fringe-pattern analysis and its applications to precision interferometry and profilometry: An overview,” Industrial Metrology 1(2), 79–99 (1990). [CrossRef]

2. Q. Kemao, “Windowed Fourier Transform for Fringe Pattern Analysis,” Appl. Opt. 43(13), 2695–2702 (2004). [CrossRef] [PubMed]

3. J. Zhong and J. Weng, “Phase retrieval of optical fringe patterns from the ridge of a wavelet transform,” Opt. Lett. 30(19), 2560–2562 (2005). [CrossRef] [PubMed]

4. I. Yamaguchi and T. Zhang, “Phase-shifting digital holography,” Opt. Lett. 22(16), 1268–1270 (1997). [CrossRef] [PubMed]

5. Z. Zhang and J. Zhong, “Applicability analysis of wavelet-transform profilometry,” Opt. Express 21(16), 18777–18796 (2013). [CrossRef] [PubMed]

6. 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]

7. M. Kujawinska and J. Wojciak, “Spatial-carrier phase-shifting technique of fringe pattern analysis,” Proc. SPIE 1508, 61–67 (1991). [CrossRef]

8. P. Chan, P. Bryanston-Cross, and S. Parker, “Spatial phase stepping method of fringe-pattern analysis,” Opt. Lasers Eng. 23(5), 343–354 (1995). [CrossRef]

9. Y. Du, G. Feng, H. Li, J. Vargas, and S. Zhou, “Spatial carrier phase-shifting algorithm based on principal component analysis method,” Opt. Express 20(15), 16471–16479 (2012). [CrossRef]

10. Y. Awatsuji, M. Sasada, and T. Kubota, “Parallel quasi-phase-shifting digital holography,” Appl. Phys. Lett. 85(6), 1069 (2004). [CrossRef]

11. J. Ville, “Theory and applications of the notion of complex signal,” RAND Corp., Santa Monica, CA, Tech. Rep. T-92 (1958).

12. H. Kwok and D. Jones, “Improved instantaneous frequency estimation using an adaptive short-time Fourier transform,” IEEE Trans. Signal Process. 48(10), 2964–2972 (2000). [CrossRef]

13. I. Daubechies, “The wavelet transform, time-frequency localization and signal analysis,” IEEE Trans. Inf. Theory 36(5), 961–1005 (1990). [CrossRef]

14. Z. Peng, G. Meng, F. Chu, Z. Lang, W. Zhang, and Y. Yang, “Polynomial chirplet transform with application to instantaneous frequency estimation,” IEEE Trans. Instrum. Meas. 60(9), 3222–3229 (2011). [CrossRef]

15. P. O'Shea and B. Boashash, “Instantaneous frequency estimation using the cross Wigner-Ville distribution with application to nonstationary transient detection,” Acoustics, Speech, and Signal Processing 5, 2887–2890 (1990). [CrossRef]

16. L. R. Watkins, S. M. Tan, and T. H. Barnes, “Determination of interferometer phase distributions by use of wavelets,” Opt. Lett. 24(13), 905–907 (1999). [CrossRef] [PubMed]

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

18. A. Asundi and Z. Wensen, “Fast phase-unwrapping algorithm based on a gray-scale mask and flood fill,” Appl. Opt. 37(23), 5416–5420 (1998). [CrossRef] [PubMed]

Spatial quasi-phase-shifting technique for single-frame dynamic fringe analysis

Abstract

1. Introduction

2. Theory

2.1 Frequency-modulated representation of fringe signal

2.2 Spatial quasi-phase-shifting technique

3. Numerical simulation

4. Experiment

4.1 Static measurement

4.2 Dynamic measurement

5. Conclusion

Acknowledgment

References and links

Supplementary Material (1)

Cited By

Figures (11)

Equations (9)

Optics Express