1. Introduction

Phase shifting interferometry (PSI) [1], exhibiting high accuracy, full field and non-contact, has been widely used in surface testing of objects. However, the phase ambiguity problem appears when the optical path difference between the adjacent pixels of the measured object is larger than half wavelength of the illumination laser. To solve this problem, dual-wavelength interferometry (DWI) [2–5] or multi-wavelength interferometry (MWI) [6,7] has been proposed, in which the phase of synthetic wavelength can be obtained by a simple subtraction between the wrapped phases of single-wavelength [8]. That is to say, the wrapped phase retrieval of single-wavelength is an important research content in DWI or MWI. In recent years, a lot of phase retrieval algorithms of single-wavelength in DWI or MWI are reported [9–16], including the spatial Fourier transform (SFT) algorithm [9,10], temporal Fourier transform (TFT) algorithm [11] and temporal phase-shifting interferometry method [12–16]. In SFT based DWI [9,10], the spatial modulations of two wavelengths are attached to one interferogram, and the wrapped phases of single-wavelength are obtained with SFT based filter processing. Though this SFT algorithm is suitable for dynamic measurements, but the corresponding space-bandwidth product of object information will be restricted and the measuring accuracy is greatly be affected by the external noise and filtering window. In TFT based DWI [11], along with the reference mirror moving with a uniform velocity, a sequence of simultaneous phase-shifting dual-wavelength interferograms (SPSDWIs) are recorded by a monochrome CCD, and then the wrapped phases of single-wavelength can be retrieved point by point through using TFT. Though there is no requirement for the fringe number in interferogram and the disturbance of external noise can be restrained effectively, but this TFT method is very time-consuming, moreover, it is required that the moving distance of the reference mirror and the number of phase-shifting interferograms should be large enough to ensure the spectral peak separation. Subsequently, the temporal phase-shifting method is introduced into DWI. In [12], a approach of single-wavelength phase-shifting (SWPS) based DWI is proposed, in which for each wavelength, a sequence of phase-shifting interferograms need to be captured. Though this SWPS method can achieve high accuracy phase of synthetic wavelength, but its recording process is greatly time-consuming and complicated. To address this, a simultaneous phase-shifting multi-wavelength interferometry based on a color CCD recording is reported [13], in which a sequence of simultaneous phase-shifting three-wavelength interferograms are captured by a color CCD, and the phase-shifting interferograms of single-wavelength are extracted through using the color separation method. Then the wrapped phases of single-wavelength can be calculated easily, but it cannot work well if the requirement of color separation is not satisfied. Additionally, several SPSDWI methods recorded with a monochrome CCD have been proposed. In [14], a two-step demodulation algorithm based SPSDWI (named as TSPS) is proposed, in which though the wrapped phases of single-wavelength can be achieved from 5-frame SPSDWIs with special phase shifts, but it is required that the accuracy of phase shifts should be enough high. In [15], based on the least-squares iterative algorithm (LSIA), a new phase retrieval method is reported. Though this method can achieve high accuracy phase, it is still time-consuming and the accuracy of phase retrieval is greatly related with the preset value of phase shifts. After that, the principle component analysis (PCA) approach based SPSDWI is reported [16]. PCA approach is firstly proposed for single-wavelength phase measurement in PSI [17–21]. In [16], the wrapped phases of single-wavelength can be retrieved, but the corresponding application condition of PCA algorithm and the sign problem of wrapped phases remain unresolved. In [21], by combing PCA and advanced iterative algorithm (AIA) [22], a new single-wavelength interferometry method is proposed.

In this study, by matching the temporal phase-shifting period number and the spatial fringe number in interferogram of two wavelengths to the integers, the phase retrieval with high accuracy can be achieved through combing PCA and LSIA (named as PCA + LSIA) method in SPSDWI. Following, we will introduce the principle of the proposed method, and then gives the simulation and experimental results to verify the validity and feasibility of the proposed method.

2. Principle

In SPSDWI, if two wavelengths of illumination lasers are respectively ${\lambda }_1$ and ${\lambda }_2$ (${\lambda }_1>{\lambda }_2$), and a sequence of SPSDWIs are captured by a monochrome CCD, thus the intensity distribution of the nth phase-shifting interferogram can be described as

Following, using PCA algorithm, we perform the phase shifts extraction of SPSDWIs. Usually, one-frame interferogram can be written as a matrix of$1\times K$, and the intensity distribution of the kth pixel in nth phase-shifting interferogram can be expressed as

Thus, N-frame background-eliminated interferograms can be represented as a matrix of $N\times K$

Then, the covariance matrix of $\tilde{I}$ can be described as

As we know, using PCA algorithm, the accuracy of phase retrieval greatly depends on the background-eliminated effect. In Eq. (4), if the phase shifts ${\theta }_{{\lambda }_1}$ and ${\theta }_{{\lambda }_2}$ are uniformly distributed in the range of $[0,n_{1}2\pi ]$ and$[0,n_{2}2\pi ]$, respectively, where $n_1$ and $n_2$ are the positive integers, we can achieve accurate background. And by using the expression of ${\theta }_{{\lambda }_2}\text{=(}{\lambda }_1/{\lambda }_2){\theta }_{{\lambda }_1}\approx (n_2/n_1){\theta }_{{\lambda }_1}$, we can find the relationship ${\lambda }_1/{\lambda }_2\approx n_2/n_1$. That is to say, so long as the phase shifts of each wavelength are distributed in the integer period, the phase-shifting period number at ${\lambda }_1$ and ${\lambda }_2$ are equal to $n_1$ and $n_2$, respectively. Additionally, according to the sampling theorem of sinusoidal signal, in the case that the $n_2$ is larger than $n_1$, the number of phase-shifting interferograms should be more than $2n_2$. In this study, we assume the greatest common divisor of $n_1$ and $n_2$ is equal to 1, so the minimum number of SPSDWIs should be equal to $2n_2+1$.

Next, we will discuss the relationship between the accuracy of phase retrieval and the fringe number in interferogram. In Eq. (6), the element of C in the ith row and the jth column can be expressed as

According to the orthogonality of trigonometric function, for each wavelength, if the fringe number in interferogram is an integer, we can obtain that

Due to ${\phi }_{{\lambda }_2}=({\lambda }_1/{\lambda }_2){\phi }_{{\lambda }_1}\approx (n_2/n_1){\phi }_{{\lambda }_1}$, when the fringe number in interferogram at ${\lambda }_1$ is equal to the integer multiple of $n_1$, Eq. (10) can be satisfied, then we can get that

Therefore $C_{ij}$ can be rewritten as

In general, if the distinguished fringe number in interferogram is not less than$n_1$, Eq. (12) can be satisfied while its accuracy is lower than above result.

In this case, the covariance matrix can be obtained by

From Eq. (13), we know that the matrix C has rank four and we need to achieve its four eigenvectors. Using the orthogonality of trigonometric function, if the phase shifts ${\theta }_{{\lambda }_1,n}$ and ${\theta }_{{\lambda }_2,n}$ are uniformly distributed in the integer periods as $[0,n_{1}2\pi ]$ and $[0,n_{2}2\pi ]$, respectively, we can get

To solve this problem, we set significantly high intensity of laser ${\lambda }_1$ relative to laser ${\lambda }_2$ [16], thus ${\lambda }_{11}\approx {\lambda }_{22}>{\lambda }_{33}\approx {\lambda }_{44}$. Due to U and D are obtained by SVD algorithm, the diagonal elements of D are arrayed with the descending order way. Therefore, ${\lambda }_{11}$ and ${\lambda }_{22}$ are corresponding to the first two diagonal elements of D, and ${\lambda }_{33}$ and ${\lambda }_{44}$ are corresponding to the second two diagonal elements of D. That is to say, $U_1$ and $U_2$ are in the first two rows of U, and $U_3$ and $U_4$ are in the second two rows of U. Assuming the first row to the fourth row of U are respectively expressed as $U_1\text{'}$, $U_2\text{'}$, $U_3\text{'}$ and $U_4\text{'}$, the phase shifts of single-wavelength can be described as

Collectively, if the matching condition, in which both the temporal phase-shifting period number and the spatial fringe number in interferogram of each wavelength are the integer, can be satisfied, the high accuracy phase shifts of single-wavelength can be achieved by PCA algorithm. However, in most cases, it is not easy to determine the accurate fringe number in the interferogram, only when the fringe number in interferogram is not less than $n_1$, the approximation condition Eq. (12) is satisfied and PCA algorithm can work well. Following, using the above ${\theta }_{{\lambda }_1}$ and ${\theta }_{{\lambda }_2}$ to perform the iterative calculation by LSIA method, we will achieve high accuracy phase.

Due to$I_{n,k}=A_k\text{+}{u}_{{\lambda }_1,k}{a}_{{\lambda }_1,n}+v_{{\lambda }_1,k}{b}_{{\lambda }_1,n}+u_{{\lambda }_2,k}{a}_{{\lambda }_2,n}+v_{{\lambda }_2,k}{b}_{{\lambda }_2,n}$, if $I_{n,k}^r$ is defined as the practical intensity of the kth pixel in nth interferogram, for the kth pixel of all phase-shifting interferograms, the sum of squares of the difference between the theoretical intensity and the practical intensity can be expressed as

To achieve the minimum of$E_k$, we have that

Combining Eq. (21) and the above phase shifts ${\theta }_{{\lambda }_1}$ and ${\theta }_{{\lambda }_2}$, we can determine $u_{{\lambda }_1,k}$, $v_{{\lambda }_1,k}$, $u_{{\lambda }_2,k}$, $v_{{\lambda }_2,k}$. Thus, the wrapped phases of single-wavelength of the kth pixel can be calculated by

Subsequently, using Eqs. (20)–(22), we can achieve the wrapped phases of single-wavelength of all pixels.

Assume that $u{\text{'}}_{{\lambda }_1,n}=B_{{\lambda }_1,k}\cos {\theta }_{{\lambda }_1,n}$, $a{\text{'}}_{{\lambda }_1,k}=\cos {\phi }_{{\lambda }_1,k}$, $v{\text{'}}_{{\lambda }_1,n}=B_{{\lambda }_1,k}\sin {\theta }_{{\lambda }_1,n}$,$b{\text{'}}_{{\lambda }_1,k}=-\sin {\phi }_{{\lambda }_1,k}$, $u{\text{'}}_{{\lambda }_2,n}=B_{{\lambda }_2,k}\cos {\theta }_{{\lambda }_2,n}$, $a{\text{'}}_{{\lambda }_2,k}=\cos {\phi }_{{\lambda }_2,k}$, $v{\text{'}}_{{\lambda }_2,n}=B_{{\lambda }_2,k}\sin {\theta }_{{\lambda }_2,n}$, $b{\text{'}}_{{\lambda }_2,k}=-\sin {\phi }_{{\lambda }_2,k}$, for all pixels of interferogram, the sum of squares of difference between the theoretical intensity and the practical intensity of nth interferogram can be expressed as

To achieve the minimum of $E_n$, we have that

Because the wrapped phases ${\phi }_{{\lambda }_1}$ and ${\phi }_{{\lambda }_2}$ have been determined, so$u{\text{'}}_{{\lambda }_1,n}$, $v{\text{'}}_{{\lambda }_1,n}$, $u{\text{'}}_{{\lambda }_2,n}$, $v{\text{'}}_{{\lambda }_2,n}$ can be achieved with Eq. (24). Thus, the phase shifts of nth interferogram can be calculated by

Using Eqs. (23)–(25), we can achieve the phase shifts of phase-shifting interferograms. Here, Eqs. (20)–(25) are employed as one iterative cycle of the LSIA. In general, the more number of iterative calculations, the higher accuracy of the retrieved phase until the convergence condition is satisfied

Finally, we can achieve the phase of synthetic wavelength by

In summary, in SPSDWI, by using PCA + LSIA method, the phase retrieval with high accuracy can be achieved if the temporal and spatial hybrid matching condition can be satisfied: (1)The phase shifts at ${\lambda }_1$ are uniformly distributed in $n_1$ periods, and the fringe number in interferogram at ${\lambda }_1$ is equal to the positive integer multiple of $n_1$; (2) based on the sampling theorem of sinusoidal signal, the number of phase-shifting interferograms should be more than $2n_2$.

3. Numerical simulation

Numerical simulation is employed to verify the effectiveness of the proposed method. Assume two illumination lasers with wavelength of ${\lambda }_1\text{=}632.8\text{nm}$ and ${\lambda }_2\text{=}532\text{nm}$ are used, then ${\lambda }_1/{\lambda }_2\approx 6/5$, $n_1=5$, $n_2=6$, and the corresponding synthetic wavelength is equal to $\Lambda ={\lambda }_1{\lambda }_2/({\lambda }_1-{\lambda }_2)=3.34\mu m$. The intensity distribution of interferogram is described as Eq. (1), in which the parameters are set as following: $-1.49\text{mm}<x,y\le 1.50\text{mm}$, and the pixel number of interferogram are set as $300\times 300$ pixels, the pixel size is equal to $\Delta x,\Delta y=10\text{μm}$, and the row and column corresponding to coordinate of x, y are represented by r and c $(1\le r,c\le 300)$; $A(x,y)=120\exp \{-5\times {10}^{-6}[{(h-150)}^2+{(l-150)}^2]\}$, $B_{{\lambda }_2}(x,y)=60\exp \{-5$ $\times {10}^{-6}[{(h-150)}^2+{(l-150)}^2]\}$, $B_{{\lambda }_1}(x,y)=1.5B_{{\lambda }_2}(x,y)$; ${\phi }_{{\lambda }_1}(x,y)=W\cdot 2\pi [{(h/150-1)}^2$ $+{(l/150-1)}^2]$, ${\phi }_{{\lambda }_2}(x,y)=({\lambda }_1/{\lambda }_2){\phi }_{{\lambda }_1}(x,y)$, in which W denotes the fringe number in interferogram and it is equal to the fringe number in interferogram at ${\lambda }_1$; ${\theta }_{{\lambda }_1,n}=2\pi T(n-1)/N+E\cdot r(n)$, ${\theta }_{{\lambda }_2,n}=({\lambda }_1/{\lambda }_2){\theta }_{{\lambda }_1,n}$, T denotes the phase-shifting period number at ${\lambda }_1$ and $E\cdot r(n)$ represents the random phase-shifting error, $r(n)$ is a random number with the average value of zero, ranging from $-1$ to 1 and r(1) = 0, and E represents the amplitude of random phase-shifting error. In addition, the simulated SPSDWIs are added the zero-mean Gaussian white noise with the standard deviation of$\sigma =2$. The convergence threshold value is set as$\epsilon =0.001\text{rad}$.

Figures 1(a)-1(d) respectively present one-frame stimulated SPSDWIs, the phase of synthetic wavelength achieved with the proposed method, the theoretical phase and the difference between the theoretical phase and the achieved phase, in which the phase-shifting step is $\pi /3$ radian at ${\lambda }_1$ and the amplitude of random phase-shifting error is set as E = 0.1 rad. The phase-shifting period number at ${\lambda }_1$ is set as T = 5, and the number of SPSDWIs is set as N = 30, and the fringe number in interferogram is set as W = 5. It is found that the root-mean-square error (RMSE) between the theoretical phase and the achieved phase is 0.020 rad, and the number of iterative calculation with the proposed method is equal to 6 when the convergence condition is satisfied.

 

Fig. 1 (a) One-frame simulated SPSDWIs; (b) the phase of synthetic wavelength achieved with the proposed method; (c) the theoretical phase; (d) the difference between (b) and (c).

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Following, using the proposed method, Figs. 2(a)–2(d) respectively present the relationship between the RMSE of phase retrieval and the phase-shifting period number at${\lambda }_1$, the fringe number in interferogram, the amplitude of random phase-shifting error and the number of SPSDWIs. For comparison, the corresponding results achieved with the PCA method are shown. Similarly, Figs. 3(a)–3(d) give the corresponding results achieved with the PCA + LSIA and LSIA methods, in which the convergence threshold value of LSIA method is set the same as the PCA + LSIA method and the iteration number is not more than 30. In Figs. 2(a) and 3(a), the fringe number in interferogram is set as W = 5, so the spatial matching condition can be satisfied. In Figs. 2(b) and 3(b), the phase-shifting period number is set as $T=5$, and the temporal matching condition can satisfied. In Figs. 2(c), 2(d), 3(c) and 3(d), both the fringe number in interferogram and the phase-shifting period number are set as 5, so both the spatial and the temporal hybrid matching condition can be satisfied. Clearly, it is found that so long as the temporal and spatial hybrid matching condition can be satisfied, each method can achieve its highest accuracy, in which the PCA + LSIA and LSIA methods need the least iteration number. Moreover, we can see that the accuracy of phase retrieval achieved with the PCA + LSIA or LSIA method is greatly higher than PCA method. In all cases, the PCA + LSIA method possesses good stability. In the case that the accuracy of phase retrieval achieved with the PCA + LSIA method is the same as the LSIA method, the required iteration number in the former is greatly less than the latter.

 

Fig. 2 The relationship between the RMSE of phase retrieval with the PCA + LSIA or PCA method and (a) the phase shifting period number at ${\lambda }_1$; (b) the fringe number in interferogram; (c) the amplitude of random phase-shifting error; (d) the number of SPSDWIs.

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Fig. 3 The relationship between the RMSE of phase retrieval or iteration number with PCA + LSIA or LSIA method and (a) the phase-shifting period number at ${\lambda }_1$; (b) the fringe number in interferogram; (c) the amplitude of random phase-shifting error; (d) the number of SPSDWIs.

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Next, to compare the accuracy and processing speed of phase retrieval with different methods, Table 1 shows the RMSE, peak to valley error (PVE) and processing time of phase retrieval achieved with the PCA + LSIA, PCA, LSIA, SWPS and TSPS methods, respectively. In PCA + LSIA, PCA and LSIA methods, it is assumed that the phase-shifting period number and the fringe number in interferogram at ${\lambda }_1$ are respectively set as T = 5, W = 5, the amplitude of random phase-shifting error and the number of SPSDWIs are set as E = 0.1 rad, N = 30, respectively. In SWPS method, for each wavelength, 30-frame phase-shifting interferograms with the phase shifts uniformly distributed in the range of [0, 2π] are captured, and the wrapped phases of single-wavelength are achieved with AIA method. In TSPS method, 5-frame SPSDWIs with the special phase shifts are employed. From Table 1, we can see that by using the temporal and spatial hybrid matching condition, the RMSE of phase retrieval with the PCA + LSIA or LSIA method are nearly the same as the high accuracy SWPS method but greatly less than the PCA or TSPS method, but the processing time with the LSIA method is more than the PCA + LSIA method. Moreover, the recording procedure of phase-shifting interferograms with the PCA + LSIA method is greatly simplified relative to the SWPS method.

Table 1. RMSE, PVE, and Processing Time Achieved with Different Methods (Simulation)

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4. Experimental results

The performance of the proposed method is also verified by experimental research. First, a SPSDWI system is built, in which a He-Ne laser with wavelength ${\lambda }_1=632.8\text{nm}$ and the diode-pumped solid-state laser with wavelength ${\lambda }_2=532\text{nm}$ are employed. Thus, the corresponding synthetic wavelength is equal to$\Lambda ={\lambda }_1{\lambda }_2/({\lambda }_1\text{-}{\lambda }_2)\text{=}3.34\text{μm}$, ${\lambda }_1/{\lambda }_2\approx 6/5$ and $n_1=5$, $n_2=6$. In addition, the convergence threshold value is set as$\epsilon =0.001\text{rad}$. A sequence of SPSDWIs is captured by a monochrome CCD with size of $1280\times 1024$ pixels and the pixel size is $5.2\text{μm}\times 5.2\text{μm}$.

In the proposed method, a sequence of 30-frame SPSDWIs with size of $388\times 388$ pixels is employed, and the corresponding phase shifts are randomly distributed in the range of $[0,n_{1}2\pi ]$ at ${\lambda }_1$. The relationship of modulation amplitude at ${\lambda }_1$ and ${\lambda }_2$ can be expressed as $B_{{\lambda }_1}\approx 1.6B_{{\lambda }_2}$. Figure 4(a) shows one-frame experimental SPSDWIs, in which the fringe number is more than 5. Figure 4(b) shows the phase of synthetic wavelength achieved with the proposed method. For comparison, Table 2 also gives the RMSE, PVE and processing time of phase retrieval achieved with the proposed PCA + LSIA, PCA, LSIA and TSPS methods, in which the phase of synthetic wavelength achieved with the SWPS method is used as the reference due to its accuracy is the highest in all listed methods in Table 1.

 

Fig. 4 (a) One-frame experimental SPSDWIs; (b) the phase of synthetic wavelength achieved with the PCA + LSIA method; (c) reference phase; (d) the difference between (b) and (c).

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Table 2. RMSE, PVE, and Processing Time of Phase Retrieval Achieved with Different Methods (Experiment)

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In SWPS method, for each wavelength, 30-frame experimental phase-shifting single-wavelength interferograms with the phase shifts randomly distributed in [0, 2π] are captured, and the wrapped phases of single-wavelength are achieved with AIA method, then the phase of synthetic wavelength is shown in Fig. 4(c). In PCA and LSIA methods, 30-frame SPSDWIs are used, in which the phase-shifts are randomly distributed in $[0,n_{1}2\pi ]$ at 632.8nm. In TSPS method, 5-frame SPSDWIs with the special phase shifts are used. Figure 4(d) presents the difference between the phase achieved with the proposed PCA + LSIA method and the reference. It is found that RMSE of phase retrieval with the proposed PCA + LSIA method is 0.034, and the iteration number is equal to 6. Clearly, we can see that the experimental results are greatly consistent with the simulated results (Table 1), further indicating the outstanding performance of the PCA + LSIA based phase retrieval method.

Furthermore, a mobile phone light guide plate is employed as the measured object. Using above experimental system, a sequence of 30-frame SPSDWIs with size of $299\times 291$ pixels are captured, in which the phase shifts are randomly distributed in $[0,n_{1}2\pi ]$ at ${\lambda }_1$, and $n_1=5$. Figures 5(a)–5(d) respectively show one-frame SPSDWIs, the phase of synthetic wavelength achieved with the proposed method, the reference phase achieved with the SWPS method and the difference between the reference phase and the achieved phase. It is found that the RMSE of phase retrieval and the iteration number are equal to 0.152 rad and 6, respectively. Clearly, this result further demonstrates the good feasibility of the PCA + LSIA based phase retrieval method.

 

Fig. 5 (a) One-frame experimental SPSDWIs; (b) the phase of synthetic wavelength achieved with the PCA + LSIA method; (c) reference phase; (d) the difference between (b) and (c).

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5. Conclusion

In summary, in SPSDWI, by introducing the temporal and spatial hybrid matching condition, requiring that both the phase-shifting period number and the fringe number in interferogram of two wavelengths are the integers, the phase retrieval with high accuracy can be achieved through combining the PCA and LSIA methods. First, by using the approximate ratio of two wavelengths, we can match both the temporal phase-shifting period number and the spatial fringe number in interferogram of two wavelengths to the integers. Second, by using above temporal and spatial hybrid matching condition, the accurate phase shifts of single-wavelength can be achieved through using PCA algorithm. In general, the requirement of fringe number in interferogram can be relaxed to not less than the minimum of phase-shifting period at the longer wavelength, and the accuracy of phase shifts is still acceptable. Third, using above phase shifts to perform the iterative calculation with LSIA method, the wrapped phases of single-wavelength can be determined. Finally, the phase of synthetic wavelength can be achieved with a simple subtraction between the wrapped phases of single-wavelength. The obtained results demonstrate that by using the temporal and spatial hybrid matching condition, the PCA + LSIA based phase retrieval method possesses significant advantages in accuracy, stability and processing time.