Two-frame fringe pattern phase demodulation using Gram-Schmidt orthonormalization with least squares method

Hangying Zhang; Hong Zhao; Zixin Zhao; Yiying Zhuang; Chen Fan

doi:10.1364/OE.27.010495

1. Introduction

Phase-shifting interferometry (PSI) is a powerful tool for high-precision testing due to its high precision and non-contact properties, such as strain, temperature and surface deformation, etc., where the phase extraction is an indispensable process. In general, the traditional phase-shifting algorithms (PSA) require at least three interferograms as input, and the phase shifts between them should be accurately known in advance. However, due to unavoidable air disturbances, mechanical vibrations and even nonlinearities of the phase shifter, these make it difficult to obtain an accurate phase shift amount, causing to the final phase extraction to be inaccurate [1–3].

In the past few decades, in order to extend the range of applications of PSI, many phase extraction algorithms have been proposed to handle random phase shifts. One of the most important classes is the iterative method which can extract the tested phase distribution (which usually characterizes the shape or deformation of the object) and/or the unknown phase shift from a series of phase shifting interferograms. Its accuracy is greatly improved at the cost of consuming more time. Okada et al. [4] first proposed a least-squares-based iterative algorithm, which solves the approximate linear equations iteratively to determine phase-shift amounts and phase distributions simultaneously. In 2004, an advanced random PSA based on a least-squares iterative procedure was proposed, it copes with the limitation of the existing iterative algorithms by separating a frame-to-frame iteration from a pixel-to-pixel iteration, and it provides stable convergence and accurate phase extraction even when the phase shifts are completely random [5]. After that, Xu et al. [6] presented an advanced iterative algorithm to extract phase distribution from randomly and spatially nonuniform phase-shifted interferograms, this algorithm divides the interferograms into small blocks and retrieves local phase shifts accurately in an iterative manner. In 2013, an iterative phase-shifting algorithm based on the least-squares principle was developed to overcome the random piston and tilt wave front errors generated from the phase shifter [7]. However, all the above iterative algorithms need at least three interferograms.

The use of a smaller number of interferograms can greatly reduce hardware and environmental requirements, so many scholars have tried to solve the problem with a single frame algorithm [8–11]. However, phase reconstruction from only a single interferogram is difficult, especially for those including closed fringes. Sign ambiguity during the single-frame demodulation is one of the main problems that impede the development of single-frame interferometry. Considering that the two-frame algorithm has higher measurement accuracy than the single-frame algorithm, and more importantly, two frames are the minimum number of interferograms used to reconstruct the phase without local symbol blur. Therefore, the two-frame algorithm is considered to be a good compromise between single-frame and multi-frame algorithms. In recent years, it has attracted the continuous attention of scholars [12–17]. Among them, there is a class of algorithms that are extremely representative, that is, Vargas et al. [15] proposed a two-frame demodulation based on the Gram-Schmidt orthonormalization method by treating two fringe patterns as independent vectors. The method is fast and accurate, which has become one of the most popular phase reconstruction approaches in two-frame interferometry. In [2,18], the superiority and efficiency of the GS algorithm have been verified in detail, so it has been widely applied [19–21]. However, almost all two-frame algorithms require the background term to be subtracted by filtering before execution, and the GS algorithm is no exception. Even though most of the background terms and noise can be suppressed by the filtering operation [22–24], we find that the residual background and noise still have a significant influence on the accuracy of the extracted phase.

The residual background and noise can easily have a significant impact on the accuracy of phase retrieval. In order to overcome this problem, this paper uses GS algorithm combined with least squares algorithm (LSA) to improve the robustness of GS algorithm, and introduce an iterative algorithm to further improve the demodulation precision. In this paper, this algorithm is called GS&LSI for short. The phase was first obtained using GS method, then a refinement operation using LSI was adopted to get the final wrapped phase map. With the optimization of the least squares method, each iteration is an effective rejection of the residual background component and the noise component in the course of multiple iterations, so the demodulation result is greatly improved in many cases.

This paper will discuss this fast and accurate two-frame phase shift algorithm with unknown phase shift in detail. The principle and process of the proposed GS&LSI algorithm are described in Section 2. In Section 3, the simulation of the GS&LSI algorithm is discussed. Section 4 provides the experimental results of the proposed method for the actual fringe pattern. The conclusion is finally drawn in Section 5.

2. Principles

2.1 Principle of two-frame GS algorithm

In phase-shifting interferometry, the intensity of two-frame interferograms can be expressed as:

I_{n} (x, y) = A_{n} (x, y) + B_{n} (x, y) \cos [φ (x, y) + δ_{n}] (n = 1, 2)

where

x

and

y

are spatial coordinates and

I_{n} (x, y)

is the intensity of two interferograms,

A_{n} (x, y)

and

B_{n} (x, y)

respectively represent the background terms and the modulation amplitudes of the two interferograms,

φ (x, y)

and

δ_{n}

are the measured phase maps and the phase shift values.

In general, the background component $A_{1} (x, y)$ and $A_{2} (x, y)$ of the above two interferograms is a smoothly varying signal and can be suppressed by a high-pass filter. By suppressing the background and ignoring the noise of two interferograms, Eq. (1) becomes:

I_{n} (x, y) \approx B_{n} (x, y) \cos [φ (x, y) + δ_{n}] (n = 1, 2)

The GS operation is an orthogonalization method that implements orthogonalization of a set of vectors [25]. In the case of orthogonalizing two-frame interferograms, the process can be easily divided into three steps.

First, a regularization process is performed on one of the interferograms. Not general, $I_{1}$ in Eq. (2) is used to represent the first interferogram and standardize it:

I_{1}^{*} = I_{1} / \sqrt{〈 I_{1}, I_{1} 〉} = I_{1} / ‖ I_{1} ‖

where the spatial dependency

(x, y)

has been omitted to simplify the equations, and

\cdot^{*}

represent the normalized vector,

〈 \cdot, \cdot 〉

and

‖ \cdot ‖

represent the inner product operator and the second-order norm.

Second, same as above, used $I_{2}$ in Eq. (2) to represent the second interferogram, and $I_{2}$ is orthogonalized with respect to the $I_{1}^{*}$ and subtract its projection to get ${\hat{I}}_{2}$ :

{\hat{I}}_{2} = I_{2} - 〈 I_{2}, I_{1}^{*} 〉 \cdot I_{1}^{*}

Finally, $I_{2}^{*}$ is obtained by dividing ${\hat{I}}_{2}$ by its second-order norm as:

I_{2}^{*} = {\hat{I}}_{2} / \sqrt{〈 {\hat{I}}_{2}, {\hat{I}}_{2} 〉} = {\hat{I}}_{2} / ‖ {\hat{I}}_{2} ‖

According to the literature [15], if the fringe number in the interferogram is more than one, the following two approximations should exist:

\begin{array}{l} | \sum_{x = 1}^{N_{x}} \sum_{y = 1}^{N_{y}} \cos^{2} (φ) \cos (δ) | ≫ | \sum_{x = 1}^{N_{x}} \sum_{y = 1}^{N_{y}} \cos (φ) \sin (φ) \sin δ | \\ B_{1} / \sqrt{\sum_{x = 1}^{N_{x}} \sum_{y = 1}^{N_{y}} {(B_{1} \cos (φ))}^{2}} \approx B_{2} / \sqrt{\sum_{x = 1}^{N_{x}} \sum_{y = 1}^{N_{y}} {(B_{2} \sin (φ))}^{2}} \end{array}

Then, the quadrature fringe pattern pair can be got by Eqs. (3) and (5):

\begin{array}{l} I_{1}^{*} = B^{'} \cos (φ) \\ I_{2}^{*} = - B^{'} \sin (φ) \end{array}

where

B^{'} ≅ B_{1} / \sqrt{\sum_{x = 1}^{N_{x}} \sum_{y = 1}^{N_{y}} {(B_{1} \cos (φ))}^{2}} ≅ B_{2} / \sqrt{\sum_{x = 1}^{N_{x}} \sum_{y = 1}^{N_{y}} {(B_{2} \sin (φ))}^{2}}

.

Thus, the two orthonormalized fringe patterns with equal modulation factor can be obtained by the GS normalization process. In this way, the phase modulo 2π can be directly computed from the two orthonormal bases as:

φ = arc \tan (\frac{- I_{1}^{*}}{I_{2}^{*}})

Through the analysis of the GS algorithm, it can be easily found that the accuracy of the GS algorithm depends greatly on the accuracy of the filtering algorithm, that is, only the complete filtering of the background term can guarantee the accuracy of the algorithm. However, all filtering algorithms can only achieve background suppression, and it is difficult to completely remove the background term, resulting in inaccurate phase calculation. In addition, two approximations in Eq. (6) of the GS algorithm inevitably lead to some measurement errors. To solve these problems, the least squares algorithm is introduced here.

2.2 Principle of least squares algorithm

The LSA can determine the amount of phase shift by preset phase distribution and can be easily incorporated into the iterative algorithm. An initial phase distribution can be estimated by the GS algorithm in Section 2.1, which can be combined with the LSA in this section. Provided that the phase of the tested surface is preset as a column vector $[φ_{1}, φ_{2}, ..., φ_{N}]$ , The phase-shifted interferogram processed by pre-filtering can be redefined as:

I_{i j} = a_{i j} + b_{i j} \cos [φ_{j} + δ_{i}]

As there may be a small amount of background residue, a term $a$ has been added in the interferograms and the modulation term $b$ is consistent with the modulation term in Eq. (2). Here, the subscript $i$ denotes the number of the phase-shifted image $(i = 1, 2)$ , $j$ denotes the individual pixel in each image $(j = 1, 2, ..., N)$ , $φ_{j}$ is the phase of the pixel $j$ , and $δ_{i}$ are the phase shift of the interferogram $i$ .

Once the initial phase distribution $φ_{j}$ is estimated, the phase shift $δ_{i}$ can be easily determined by using of the least-squares iteration method. The background term changes slowly with respect to the cosine of the phase in the actual conditions. To avoid damage to the fringe information during filtering, when we select high-pass filtering, a filter with a small suppression band is usually used, and the relative fringe information of the background residual is still a low-frequency component. Especially when the fringe information contains some low-frequency fringe components, due to the existence of aliasing, this allows us to select only a small suppression band, which will cause residual of the low-frequency background term. Therefore, it is not difficult to understand that the residual background term is still low frequency after being processed by the high pass filtering method. It is assumed here that the residual background intensity and the modulation amplitude do not have pixel-to-pixel variation, i.e. $a_{i 1} = a_{i 2} = ... = a_{i N}$ and $b_{i 1} = b_{i 2} = ... = b_{i N}$ , that they are only the functions of frames. A new set of variables are defined for the i-th frame as: $a_{i}^{'} = a_{i}$ , $b_{i}^{'} = b_{i} \cos δ_{i}$ and $c_{i}^{'} = - b_{i} \sin δ_{i}$ , Eq. (9) becomes [6]:

I_{i j} = a_{i}^{'} + b_{i}^{'} \cos φ_{j} + c_{i}^{'} \sin φ_{j}

The squared sum of the differences between the theoretical intensity and actual intensity of the interferogram can be expressed as

S_{i} = \sum_{j = 1}^{N} (I_{i j}^{t} - I_{i j})^{2} = \sum_{j = 1}^{N} (a_{i}^{'} + b_{i}^{'} \cos φ_{j} + c_{i}^{'} \sin φ_{j} - I_{i j})^{2}

According to the least squares theory [5–7], $S_{i}$ should be minimum when

\partial S_{i} / \partial a_{i}^{'} = 0, \partial S_{i} / \partial b_{i}^{'} = 0, \partial S_{i} / \partial c_{i}^{'} = 0

so

{X_{i}} = {[S_{i}]}^{- 1} {R_{i}}

[\begin{matrix} a_{i}^{'} \\ b_{i}^{'} \\ c_{i}^{'} \end{matrix}] = {[\begin{matrix} \begin{matrix} N & \sum_{j = 1}^{N} \cos φ_{j} & \sum_{j = 1}^{N} \sin φ_{j} \end{matrix} \\ \begin{matrix} \sum_{j = 1}^{N} \cos φ_{j} & \sum_{j = 1}^{N} \cos^{2} φ_{i} & \sum_{j = 1}^{N} \sin φ_{j} \cos φ_{j} \end{matrix} \\ \begin{matrix} \sum_{j = 1}^{N} \sin φ_{j} & \sum_{j = 1}^{N} \sin φ_{j} \cos φ_{j} & \sum_{j = 1}^{N} \sin^{2} φ_{j} \end{matrix} \end{matrix}]}^{- 1} [\begin{matrix} \sum_{j = 1}^{N} I_{i j} \\ \sum_{j = 1}^{N} I_{i j} \cos φ_{j} \\ \sum_{j = 1}^{N} I_{i j} \sin φ_{j} \end{matrix}]

then the unknowns

a_{i}^{'}, b_{i}^{'}

and

c_{i}^{'}

can be obtained by Eq. (14), and the amount of phase shift value of the interferogram i can be determined from

δ_{i} = \tan^{- 1} (- c_{i}^{'} / b_{i}^{'})

Also, the background intensity and the modulation amplitude can be extracted from Eq. (14),

a_{i} = a_{i}^{'}, b_{i} = \sqrt{{(b_{i}^{'})}^{2} + {(c_{i}^{'})}^{2}}

.

For two phase-shifted interferograms, the phase shift $δ_{1}$ and $δ_{2}$ of these two interferograms can be calculated by Eq. (15), the relative phase shift is $δ = δ_{2} - δ_{1}$ .

Then, using Eqs. (16) and (17) to calculate the new phase distribution,

\begin{array}{l} \cos φ = \frac{I_{1} - a_{1}}{b_{1}} \\ \sin φ = \frac{(\frac{b_{2} I_{1}}{b_{1}} \cos δ - I_{2}) - (\frac{b_{2}}{b_{1}} a_{1} \cos δ - a_{2})}{b_{2} \sin δ} \end{array}

φ = \tan^{- 1} (\frac{(b_{2} I_{1} \cos δ - b_{1} I_{2}) - (b_{2} a_{1} \cos δ - b_{1} a_{2})}{(I_{1} - a_{1}) b_{2} \sin δ})

Note that the phase map obtained using Eq. (17) usually has higher precision than that using Eq. (8). As a result, a higher phase accuracy can be obtained by introducing an iterative algorithm.

2.3 Principle of GS and least squares iterative phase shifting algorithm

Based on the principle of GS and LSA, we can make full use of the advantages of the two methods by combining Gram-Schmidt orthogonal normalization and least squares algorithm. To improve the accuracy of calculation, an iteration process is introduced. And the detailed procedure of our proposed method is described as follows:

1) Apply Eq. (8) to estimate the initial phase distribution;
2) Use the initial phase distribution obtained in step 1) to calculate the relative phase shift $δ$ , the residual background intensity $a_{1}$ and $a_{2}$ , and modulation amplitude $b_{1}$ and $b_{2}$ can be obtained in the LSA, then apply Eq. (17) to calculate the new phase distribution;
3) Repeat step 2) with the new phase distribution until $R M S (φ^{k} - φ^{k - 1}) < ξ$ , the final phase distribution $φ$ and the relative phase shift $δ$ can be obtained.

Where $ξ$ is the predefined iteration converging threshold, for example, 1e-5, and k presents the number of iterations.

The whole procedure of the GS&LSI algorithm is illustrated in Fig. 1. The dotted line in Fig. 1 is the iterative operation process.

Fig. 1 Flow chart of GS&LSI phase-shifting algorithms.

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3. Simulation

To validate the effectiveness and robustness of the proposed GS&LSI method, four simulations are performed under different conditions. Considering that deleting the background term is a prerequisite for executing the GS algorithm, all filtering algorithms can only suppress the background term to the greatest extent in practice, and it is impossible to completely remove the background term. Therefore, the advantages and disadvantages of the filtering algorithms are outside the scope of this paper. In all simulations of this article, the background terms are removed directly, and only the background residuals (usually very small amounts) are analyzed. In the four simulations, the effect of different background residuals is first considered. The effects of different phase shifts, different fringe numbers and different random noise are studied separately next. The reference phase distribution (512 pixels*512 pixels) is simulated firstly with $φ = 5 p e a k s + (50 x^{2} - 200 x - 10 y^{2} + 400 y) / 20$ as shown in Fig. 2(a), and then Eq. (9) is used to generate two interferograms by setting a random phase shift value (e.g., 1.2515 rad) between them (Figs. 2(b) and 2(c)).

Fig. 2 Simulated phase distribution and the two phase-shifting interferograms. (a) The reference phase, (b) the first interferogram, and (c) the second interferogram.

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In the first simulation, the effects of different background residuals on the GS method and the proposed method are investigated. To ensure generality, uniform background residuals and non-uniform background residuals are discussed.

Considering that different background residuals will produce different effects, we set four different sizes of uniform background residuals (a = 0, a = 0.02, a = 0.05, a = 0.10). Since the modulation term in the simulation does not affect the measurement result, it is uniformly set to a relatively large value (b = 1.24). The residual maps, that is the difference between the calculated phase and the reference phase, obtained by the two methods are given in Fig. 3.

Fig. 3 The residual maps obtained by the GS method in the uniform background residuals with different size: (a) a = 0;(b) a = 0.02;(c) a = 0.05;(d) a = 0.10; and the residual maps obtained by the GS&LSI method in the uniform background residuals with different size:(e) a = 0; (f) a = 0.02; (g) a = 0.05; (h) a = 0.10.

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By comparing Figs. 3(a)-(d) with Figs. 3(e)-(h), we find that the GS&LSI algorithm can effectively suppress the influence of background residual, and the obtained phase distribution map is extremely close to the reference phase. The residual error is corrected to a very small order of magnitude (1e-6), while the residual error is very large for the GS method, even if the background term is completely rejected. It shows that the GS&LSI algorithm has excellent error correction capability for the case of uniform background residuals.

To investigate the performance of our proposed method under non-uniform background residuals condition, the Gaussian distribution was added to each background residual amount. By using the Gaussian parameter to set background residuals size, we use the parameters (σ = 30, σ = 20, σ = 8, σ = 4) for four non-uniform cases. The setting of the modulation term is consistent with the case of uniform residuals. Figure 4 shows the corresponding residual maps obtained using the two methods. As can be seen in Figs. 4(e) and 4(f), the proposed method can also correct the phase error well in the case of relatively small non-uniform background residuals. As the background residual increases, which is shown in Figs. 4(g) and 4(h), the performance of our proposed algorithm is not so satisfactory. The reason might be that the uniform background assumption in the LSI process doesn’t hold true when the background residual fluctuation is relatively large. Although it can't correct all the residual points to a small value, by comparing Figs. 3(c) and 3(d), the number of residual points obtained by the proposed algorithm is greatly reduced, which proves that the GS&LSI algorithm still exhibits strong error correction capability.

Fig. 4 The residual maps obtained by the GS method in the non-uniform background residuals with different Gaussian parameter: (a) σ = 30;(b) σ = 20;(c) σ = 8;(d) σ = 4; and the residual maps obtained by the GS&LSI method in the non-uniform background residuals with different Gaussian parameter: (e) σ = 30; (f) σ = 20; (g) σ = 8; (h) σ = 4.

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Table 1 presents the RMS value of the residual maps and the consumed time for the GS and our GS&LSI method. Regardless of the uniform background residual or the non-uniform background residual, the error of the GS method increases significantly as the background residual increases. At this time, the corresponding error of the GS&LSI method is far less than the error of the GS method, which indicates that the GS&LSI algorithm can effectively suppress the influence of the background residual on the phase demodulation. By analyzing the case where the background residual is 0, it can be found that the GS&LSI method is not only insensitive to background residual, but also has a good inhibitory effect on the approximation of the GS method.

Table 1. The demodulation errors and consumed time for GS and GS&LSI in different background residuals

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Table 1 also shows the processing time for different algorithms. As can be seen from Table 1, the GS&LSI algorithm has similar processing times in eight different cases, indicating that different background intensity residuals do not affect processing time. Since LSI requires more time, the GS&LSI has a longer processing time than the GS. In addition, it can be seen in Fig. 5 that the phase measurement accuracy is multiplied after only one run of the least-squares operation, and the time taken by the proposed algorithm even after ten iterations is still much less than one second. Therefore, the proposed algorithm has great advantages on the demodulation accuracy.

Fig. 5 The iterative demodulation RMS error curve (rad) for GS&LSI in the uniform background residuals with different size: (a) a = 0;(b) a = 0.02;(c) a = 0.05;(d) a = 0.10; and the non-uniform background residuals with different Gaussian parameter: (e) σ = 30;(f) σ = 20;(g) σ = 8;(h) σ = 4.

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The amount of phase shift between frames can also be calculated by the least-squares method. Therefore, for the GS&LSI algorithm, the phase shift error is calculated here. It shows that the phase shift values calculated by GS&LSI are very close to the preset phase shift amount (1.2515 rad), which have been shown in Table 2. This result further proves that the GS&LSI algorithm has extremely high measurement accuracy.

Table 2. The phase shift errors of GS&LSI in the different background residuals

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Since the phase shift value between frames is important for the two-frame phase shift algorithm, it is necessary to discuss the robustness of the GS&LSI algorithm for different phase shift values between different frames. In the second simulation, by uniformly arranging the distribution of the phase shift values in the range of 0.1 rad to 2.9 rad, the other simulation conditions are the same as the cases in the first simulation, respectively. Under normal circumstances, the range should be 0 to π, 0 rad is not selected because it means no phase shift, and here 2.9 rad is chosen as the maximum phase shift value since the phase error is too large when the phase shift value is more than 2.9 rad for the GS algorithm [26]. Figure 6 shows the phase measurement error at different phase shift values in several cases. It can be seen from the figure that the GS algorithm is affected by the residual of the background intensity, and a relatively large phase error is generated under almost all phase shift values. Especially when the phase shift exceeds 2.5 rad, the phase measurement error increases significantly. In addition, even if the background residual is completely removed (Fig. 6(a)), the GS&LSI algorithm can achieve higher measurement accuracy because it can effectively suppress the approximation operation in the GS algorithm.

Fig. 6 The effect of the amount of phase shift between frames on the measurement results of the two algorithms under uniform background residuals with different sizes: (a) a = 0;(b) a = 0.02;(c) a = 0.05; (d) a = 0.10; and under non-uniform background residuals with different Gaussian parameter: (e) σ = 30;(f) σ = 20;(g) σ = 8;(h) σ = 4.

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In the third simulation, the measurement accuracy of the GS and GS&LSI algorithms is compared by setting different number of fringes. Due to the multiple approximation operations in the GS algorithm, the number of fringes in the interferogram also has some impact on the accuracy of the phase measurement. In order to avoid the influence of other factors, the case where the background term is completely removed is selected. According to the curve analysis in Fig. 7(a), as the number of fringes increases, the measurement accuracy of the GS algorithm gradually increases, indicating that the approximation in the GS algorithm is closer to the actual data when there are many fringes. However, when the number of fringes is small, the approximation operation still introduces certain measurement errors. Compared with the GS&LSI algorithm, even if the number of fringes is small, the proposed algorithm still maintains extremely high measurement accuracy. The results show that the GS&LSI algorithm can effectively repair the measurement error introduced by the approximation operation, especially when the number of fringes is small, the repair effect is more obvious.

Fig. 7 The phase demodulation errors (RMS) of GS and GS&LSI for (a) different fringe number and (b) different random noise.

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The residual of the background term still belongs to the DC component. Therefore, in the fourth simulation, it is necessary to consider the influence of random noise on the accuracy of phase measurement. In this simulation, the random noise is added to the two interferograms using the awgn function in MATLAB. The phase error of the different noises is analyzed from 10 dB to 60 dB with an interval of 5 dB. The results are shown in Fig. 7(b). The phase error decreases as the SNR increases from 5 dB to 60 dB. When the SNR is greater than 50 dB, the phase error is negligible. In addition, the GS is more sensitive to random noise than the GS&LSI. Figure 7(b) also shows that GS&LSI can better suppress random noise with a phase error approximately equal to zero. In order to ensure the accuracy of the measurement, the reliability of the above data is ensured by averaging of four measurements.

The four numerical simulations indicate that the proposed algorithm is robust in different situations. The GS&LSI algorithm could extract the phase with high precision from the interferograms with an unknown phase shift. Based on the above different simulations, the advantages of the proposed GS&LSI algorithm can be summarized as: 1) It is almost insensitive to the residual background intensity; 2) it is less sensitive to the inter-frame phase shift value; 3) it can effectively repair the approximation error introduced when the number of fringes is small; 4) it can partially suppress the effect of measurement errors introduced by random noise. In short, the proposed algorithm can effectively improve the measurement accuracy of the GS algorithm.

4. Experimental verification

Experiments were also conducted to demonstrate the effectiveness of the proposed algorithm. Without loss of generality, two sets of fringe patterns were collected in the experiment, one with an open fringes pattern and the other with a closed fringes pattern. Here we use the measurement results of the advanced spatial spectrum fitting algorithm (ASSF), which is a novel four-step random phase shift method, as the reference phase [27]. First, four phase shifting fringe patterns were acquired by the phase shift method and two frames are selected as the object of the two-frame phase shift algorithm. To ensure the validity of the experimental results, a Gaussian high-pass filtering algorithm is used to suppress the background term and the noise term. In [8, two sets of real interference images and the measurement results of different demodulation methods are shown, respectively, where Figs. 8(a)-(e) are the case of real open fringes, and the case of real closed fringes are presented in Figs. 8(f)-(j).

Fig. 8 The real open fringes (a) first interferogram; (b) second interferogram; the tested phase by different algorithms (c) ASSF; (d) GS; (e) GS&LSI; and the real closed fringes (f) first interferogram; (g) second interferogram; the tested phase by different algorithms (h) ASSF; (i) GS; (j) GS&LSI.

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The RMS errors of the phase distribution obtained by the GS and GS&LSI methods is shown in Table 3, respectively. For the GS method, the RMS error values for both open and closed fringes (0.3607 rad and 0.8372 rad, respectively) indicate a large difference between the calculated phase and the reference phase. The main reason is that the GS algorithm is sensitive to the residual background intensity, and it can be noted that when the fringes in the interferogram are small, the residual error is large. In contrast, due to the iterative correction process, the RMS values of the GS&LSI method become very small (0.1859 rad and 0.2425 rad, respectively). It is shown that the surface shape calculated by the proposed method is closer to the result of the ASSF method.

Table 3. The demodulation RMS errors of GS and GS&LSI for two sets of real fringes

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Figure 9 shows the relationship between the calculated phase shift value with the number of iterations and the relationship between the residual RMS error with the number of iterations. It can be seen that the phase shift values remain almost unchanged after several iterations, which indicates that our proposed method can get very accurate phase shift amount between the two interferograms. As a result, the phase measurement accuracy can be effectively improved without sacrificing too much time.

Fig. 9 The phase shift value with iteration (a) the open fringes;(b) the closed fringes; and the RMS error with iteration (c) the open fringes; (d) the closed fringes.

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To verify the robustness of GS&LSI, the complex fringe patterns are also discussed, and the results of the two different methods are compared. Figures 10(a) and 10(b) show the two-frame phase shifting interferograms, and Figs. 10(c)-10(d) show the phase distribution extracted using the ASSF, GS&LSI and GS methods, respectively. Similar to the previous experiment, the phase distribution extracted by the ASSF is set as the reference phase distribution. Table 4 shows the difference between the tested phase distribution and the reference phase extracted by GS&LSI and GS. The RMS values are 0.8812 rad and 0.4394 rad, respectively. We can see that the phase retrieval accuracy of GS&LSI for complex fringes is also higher than that of GS phase retrieval.

Fig. 10 The complex fringes (a) first interferogram; (b) second interferogram; the tested phase by different algorithms (c) ASSF; (d) GS; (e) GS&LSI.

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Table 4. The demodulation RMS errors of GS and GS&LSI for complex fringes

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Through the above experiments, we verified that for simple and complex fringes, it can be shown that: 1) the proposed GS&LSI can obtain a relatively accurate result by only two interferograms, and it can partially suppress the effect introduced the residual background intensity; 2) The proposed GS&LSI method has a stable convergence speed. It demonstrates that the algorithm has good efficiency while ensuring a higher measurement accuracy.

5. Conclusion

In this paper, we present a random two-frame phase shifting algorithm based on Gram-Schmidt orthonormalization and least squares technologies. Firstly, the initial phase distribution is calculated by the standard GS phase shift algorithm, and then the optimization of the least squares iterative method is used to extract a more accurate phase distribution and phase shift. The proposed algorithm can achieve higher precision than the standard GS and can be used under a variety of complex conditions, such as a working environment where background terms cannot be completely removed, and interference patterns with few fringes. The proposed method is proved by simulation data and experimental data, and it has potential application for high-precision phase extraction in two-frame phase shift interferometry.

Funding

National Key Research and Development Programs of China (2018YFC1603500); National Natural Science Foundation of China (NSFC) (61575157, 51705404); China Postdoctoral Science Foundation (2016M602806); Fundamental Research Funds for the Central Universities (xjj2017093); China Scholarship Council Foundation (201806285004).

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	methods	uniform background residuals				non-uniform background residuals
	methods	a = 0	a = 0.02	a = 0.05	a = 0.10	σ = 30	σ = 20	σ = 88	σ = 48
RMS error (rad)	GS	0.0043	0.0147	0.0354	0.0706	0.0102	0.0146	0.0351	0.0689
RMS error (rad)	GS&LSI	4.3e-6	3.9e-6	4.5e-6	4.7e-6	5.9e-6	1.1e-5	1.2e-4	9.1e-4
Processing time(s)	GS	0.0195	0.0175	0.0171	0.0183	0.0166	0.0173	0.0180	0.0172
Processing time(s)	GS&LSI	0.3217	0.3539	0.3813	0.3884	0.3127	0.3368	0.3277	0.3827

	methods	uniform background residuals				non-uniform background residuals
	methods	a = 0	a = 0.02	a = 0.05	a = 0.10	σ = 30	σ = 20	σ = 88	σ = 48
RMS error (rad)	GS	0.0043	0.0147	0.0354	0.0706	0.0102	0.0146	0.0351	0.0689
RMS error (rad)	GS&LSI	4.3e-6	3.9e-6	4.5e-6	4.7e-6	5.9e-6	1.1e-5	1.2e-4	9.1e-4
Processing time(s)	GS	0.0195	0.0175	0.0171	0.0183	0.0166	0.0173	0.0180	0.0172
Processing time(s)	GS&LSI	0.3217	0.3539	0.3813	0.3884	0.3127	0.3368	0.3277	0.3827

Two-frame fringe pattern phase demodulation using Gram-Schmidt orthonormalization with least squares method

Abstract

1. Introduction

2. Principles

2.1 Principle of two-frame GS algorithm

2.2 Principle of least squares algorithm

2.3 Principle of GS and least squares iterative phase shifting algorithm

3. Simulation

4. Experimental verification

5. Conclusion

Funding

References

Cited By

Figures (10)

Tables (4)

Equations (17)

Optics Express

	methods	Open fringes	Close fringes
RMS error (rad)	GS	0.3607	0.8372
RMS error (rad)	GS&LSI	0.1859	0.2425