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Rapid and precise phase retrieval from two-frame tilt-shift based on Lissajous ellipse fitting and ellipse standardization

Yanping Fu, Qianchao Wu, Yong Yao, Yaping Gan, Chuyan Liu, Yanfu Yang, Jiajun Tian, and Ke Xu

Open Access Open Access

Abstract

A rapid and precise phase-retrieval method based on Lissajous ellipse fitting and ellipse standardization is demonstrated. It only requires two interferograms without pre-filtering, which reduces its complexity and shortens the processing time. The elliptic coefficients obtained by ellipse fitting are used for ellipse standardization. After compensating phase-shift errors by ellipse standardization, the phase distribution is extracted with high precision. It is suitable for fluctuation, noise, tilt-shift, simple and complex fringes. This method is effective for the number of fringes less than 1. The reliability of the method is verified by simulations and experiments, indicating high accuracy and less time consumption.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1. Introduction

Phase-shifting interferometry (PSI) with non-contact and high precision is a powerful technology for optical measurement [1], optical imaging [2], digital holography [3], etc. In PSI, a crucial technique is phase shifting algorithm (PSA). The classic PSA needs to know the determined phase-shift value [4]. However, the accuracy of the phase extraction drops sharply due to the phase-shift errors [5]. The phase-shift errors including spatially non-uniformity, disturbance, translational-shift and tilt-shift are caused by the instability of light source, air turbulence, miss-calibration and mechanical vibration [6–9], respectively. Substantial efforts have been made to compensate the translational-shift, including the principal component analysis (PCA) [10], the Lissajous figure and ellipse fitting (LEF) [11], the Gram-Schmidt orthogonal normalization and Lissajous ellipse fitting (GS&LEF) [12], the Gram-Schmidt orthogonal normalization and least squares iterative (GS&LSI) [13], the principle component analysis and Lissajous ellipse fitting (PCA&LEF) [14], the Lissajous ellipse fitting and least squares iterative (LEF&LSI) [15], etc. However, it’s difficult to maintain a constant phase-shift value among different pixels due to nonlinearity and spatially non-uniformity of the phase shifter [16]. In 2008, Xu et al. proposed the advanced iterative algorithm to extract tilt-shift [17], but this algorithm is time consuming and can only deal with the small amplitude tilt-shift error (within 27.6%). To improve the flexibility of phase extraction, in 2013, Liu et al. proposed a three-step least squares iterative phase extraction algorithm [18]. Although this algorithm can process large amplitude tilt-shift, it is time consuming. Recently, Wielgus et al. use nonlinear error functional minimization to extract tested phase from two interferograms for fast dynamic measurement [19]. However, this algorithm needs filtering out the background intensity in advance, cannot extracts phase well when there is nonlinear tilt-shift. Hence, we propose an efficient phase extraction algorithm that is insensitive to linear and nonlinear tilt-shift without pre-filtering.

In this paper, we present a rapid and precise phase extraction algorithm from two interferograms based on Lissajous ellipse fitting and ellipse standardization. This method is insensitive to linear and nonlinear tilt-shift and does not need filtering out the background intensity. The Lissajous ellipse fitting is used to extract phase distribution from two interferograms where these interferograms suffer from phase errors including noise, disturbance, spatially non-uniform and tilt-shift. It can acquire high precision because phase-shift errors are compensated successfully by transforming oblique-ellipse into standard-ellipse. Section 2 presents the principle and process of the proposed algorithm based on ellipse fitting and ellipse standardization. In Section 3, the simulation of proposed algorithm is discussed. In Section 4, the proposed algorithm is evaluated with the experimental data. Finally, the conclusion is illustrated in Section 5.

2. Principle

In PSI, the general two phase shifted interferograms can be expressed as:

$${I_1}(x,y) = {a_1}(x,y) + {b_1}(x,y)\cdot \cos [\phi (x,y)]$$
$${I_2}(x,y) = {a_2}(x,y) + {b_2}(x,y)\cdot \cos [\phi (x,y) + \delta (x,y)]$$
where, $x$ and y are the pixel positions of rows and columns, ${I_1}(x,y)$ and ${I_2}(x,y)$ are the intensity of two interferograms, ${a_1}(x,y)$ and ${a_2}(x,y)$ represent the background intensity, ${b_1}(x,y)$ and ${b_2}(x,y)$ represent the modulation amplitude, $\phi (x,y)$ represents the tested phase, $\delta (x,y)$ represents the phase-shift value between two interferograms.

For simplicity, we omit the $(x,y)$ in following discussion, then, Eqs. (1) and (2) can be rewritten as:

$${I_1} = {a_1} + {b_1}\cdot \cos \phi $$
$${I_2} = {a_2} + {b_2}\cdot \cos (\phi + \delta ) = {a_2} + {b_2}\cdot \cos \phi \cdot \cos \delta - {b_2}\cdot \sin \phi \cdot \sin \delta $$
From Eqs. (3) and (4) we can obtain:
$$\cos \phi = ({I_1} - {a_1})/{b_1}$$
$$\sin \phi = \frac{{{a_2}}}{{{b_2}\cdot \sin \delta }} - \frac{{{a_1}\cdot \cos \delta }}{{{b_1}\cdot \sin \delta }} + \frac{{\cos \delta }}{{{b_1}\cdot \sin \delta }}\cdot {I_1} - \frac{1}{{{b_2}\cdot \sin \delta }}\cdot {I_2}$$
Because ${\sin ^2}\phi + {\cos ^2}\phi = 1$, Eqs. (5) and (6) can be rewritten as:
$$\begin{array}{l} \frac{1}{{{b_1}\cdot {{\sin }^2}\delta }}\cdot I_1^2 - \frac{{2\cdot \cos \delta }}{{{b_1}\cdot {b_2}\cdot {{\sin }^2}\delta }}\cdot {I_1}\cdot {I_2} + \frac{1}{{b_2^2\cdot {{\sin }^2}\delta }}\cdot I_2^2\textrm{ + (}\frac{{2\cdot {a_2}\cdot \cos \delta }}{{{b_1}\cdot {b_2}\cdot {{\sin }^2}\delta }} - \frac{{2\cdot {a_1}\cdot {{\cos }^2}\delta }}{{b_1^2\cdot {{\sin }^2}\delta }} - \frac{{2\cdot {a_1}}}{{b_1^2}})\cdot {I_1}\\ + (\frac{{2\cdot {a_1}\cdot \cos \delta }}{{{b_1}\cdot {b_2}\cdot {{\sin }^2}\delta }} - \frac{{2\cdot {a_2}}}{{b_2^2\cdot {{\sin }^2}\delta }})\cdot {I_2} + {(\frac{{{a_2}}}{{{b_2}\cdot \sin \delta }} - \frac{{{a_1}\cdot \cos \delta }}{{{b_1}\cdot \sin \delta }})^2} + \frac{{a_1^2}}{{b_1^2}} - 1 = 0 \end{array}$$
Equation (7) can be expressed as an ellipse form:
$$a\cdot {I_1}^2 + b\cdot {I_1}\cdot {I_2} + c\cdot {I_2}^2 + d\cdot {I_1} + f\cdot {I_2} + g = 0$$
where, $a,b,c,d,f,g$ are the coefficients of the ellipse, $a = \frac{1}{{{b_1}\cdot {{\sin }^2}\delta }}$, $b ={-} \frac{{2\cdot \cos \delta }}{{{b_1}\cdot {b_2}\cdot {{\sin }^2}\delta }}$, $c = \frac{1}{{b_2^2\cdot {{\sin }^2}\delta }}$, $d = \frac{{2\cdot {a_2}\cdot \cos \delta }}{{{b_1}\cdot {b_2}\cdot {{\sin }^2}\delta }} - \frac{{2\cdot {a_1}\cdot {{\cos }^2}\delta }}{{b_1^2\cdot {{\sin }^2}\delta }} - \frac{{2\cdot {a_1}}}{{b_1^2}}$, $f = \frac{{2\cdot {a_1}\cdot \cos \delta }}{{{b_1}\cdot {b_2}\cdot {{\sin }^2}\delta }} - \frac{{2\cdot {a_2}}}{{b_2^2\cdot {{\sin }^2}\delta }}$, $g = {(\frac{{{a_2}}}{{{b_2}\cdot \sin \delta }} - \frac{{{a_1}\cdot \cos \delta }}{{{b_1}\cdot \sin \delta }})^2} + \frac{{a_1^2}}{{b_1^2}} - 1$.

Equation (8) can be also expressed as follow:

$$\frac{{{{(({I_1} - {x_0})\cdot \cos \theta + ({I_2} - {y_0})\cdot \sin \theta )}^2}}}{{a_x^2}} + \frac{{{{(({I_2} - {y_0})\cdot \cos \theta - ({I_1} - {x_0})\cdot \sin \theta )}^2}}}{{a_y^2}} - 1 = 0$$
where, ${x_0},{y_0},{a_x},{a_y},\theta $ are the parameters of the ellipse, ${x_0}$ and ${y_0}$ represent the center offset, ${a_x}$ represents the semi-major amplitude, ${a_y}$ represents the semi-minor amplitude, $\theta $ represents the ellipse orientation angle with respect to the x-axis.

we can get the conic-to-parametric transformation parameters as follow:

$$\begin{aligned} {x_0} &= \frac{{b\cdot f - 2c\cdot d}}{{4\cdot a\cdot c - {b^2}}},{y_0} = \frac{{b\cdot d - 2a\cdot f}}{{4\cdot a\cdot c - {b^2}}}\\ {a_x} &= \sqrt {2\cdot \frac{{a\cdot {f^2} + c\cdot {d^2} + g\cdot {b^2} - b\cdot d\cdot f - 4\cdot a\cdot c\cdot g}}{{({b^2} - 4\cdot a\cdot c)(\sqrt {{{(a - c)}^2} + {b^2}} - (a + c))}}} \\ {a_y} &= \sqrt {2\cdot \frac{{a\cdot {f^2} + c\cdot {d^2} + g\cdot {b^2} - b\cdot d\cdot f - 4\cdot a\cdot c\cdot g}}{{({b^2} - 4\cdot a\cdot c)( - \sqrt {{{(a - c)}^2} + {b^2}} - (a + c))}}} \\ \theta &= \left\{ \begin{array}{lllll} 0 & {for}& {b = 0}&{and}&{a\;<\;c}\\ \frac{\pi }{2} &{for} &{b = 0}&{and}&{a\;>\;c} \\ \frac{1}{2}\cdot \arctan (\frac{b}{{a - c}}) &{for}&{b \ne 0} &{and}&{a\;<\;c} \\ \frac{\pi }{2} + \frac{1}{2}\cdot \arctan (\frac{b}{{a - c}})&{for} &{b \ne 0}&{and} &{a\;>\;c} \end{array}\right. \end{aligned}$$
Equation (9) can be rewritten as follow:
$$\left\{ \begin{array}{l} \sin \phi = \frac{X}{{{a_x}}}\\ \cos \phi = \frac{Y}{{{a_y}}} \end{array} \right.$$
where, $X = ({I_1} - {x_0})\cdot \cos \theta + ({I_2} - {y_0})\cdot \sin \theta $, $Y = ({I_2} - {y_0})\cdot \cos \theta - ({I_1} - {x_0})\cdot \sin \theta $.

Finally, from Eqs. (10) and (11), the calibrated phase is given in follow equation.

$$\phi = \arctan (r\cdot \frac{X}{Y})\textrm{ = }\arctan (r\cdot \frac{{({I_1} - {x_0})\cdot \cos \theta + ({I_2} - {y_0})\cdot \sin \theta }}{{({I_2} - {y_0})\cdot \cos \theta - ({I_1} - {x_0})\cdot \sin \theta }})$$
where, $r = {a_y}/{a_x}$.

At first, we plot the Fig. 1 to illustrate how the proposed method robust against phase-shift errors including spatially non-uniformity, fluctuation and tilt-shift. According to Eqs. (1) and (2), we draw ${I_1}$ as the horizontal coordinate and ${I_2}$ as the vertical coordinate to acquire an oblique ellipse with errors as shown in Fig. 1(a). The errors of spatially non-uniformity, fluctuation and tilt-shift are caused by the instability of light source, environmental disturbance and mechanical vibration, respectively. Then, we use the least squares fitting to obtain the coefficients of the oblique ellipse, after that, the transformation parameters are calculated by Eq. (10). The standard ellipse without errors is shown in Fig. 1(b). The phase-shift errors are compensated successfully by transforming oblique-ellipse into standard-ellipse with transformation parameters. Finally, the real phase can be obtained by Eq. (12).

 figure: Fig. 1.

Fig. 1. Lissajous figures. (a) The oblique ellipse (with errors), (b) the standard ellipse (without errors).

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3. Simulation

The robustness and feasibility of the proposed algorithm are discussed from six different conditions, including tilt-shift, noise, fluctuation, different number of fringes, complex fringes and highly non-uniformity. All computations are performed using a processor of Inter I3 3.4 GHz. In the following simulations, the proposed algorithm is compared with Wielgus’s algorithm under the condition of tilt-shift because Wielgus’s algorithm has high accuracy with linear tilt-shift. The image size is 512*512.

3.1 Performance comparison between linear tilt-shift and nonlinear tilt-shift

First, we consider small disturbance of the background intensity and the modulation amplitude. These parameters are set as:

$$\begin{array}{l} {a_1}({x,y} )= 1 + 0.1\cdot exp({ - 0.02({{x^2} + {y^2}} )} ),\ {a_2}({x,y} )= 0.9 + 0.21\cdot exp({ - 0.02({{x^2} + {y^2}} )} )\\ {b_1}({x,y} )= 1 + 0.1\cdot exp({ - 0.02({{x^2} + {y^2}} )} ),\ {b_2}({x,y} )= 0.9 + 0.21\cdot exp({ - 0.02({{x^2} + {y^2}} )} )\end{array}$$
The phase distribution is set as:$\phi (x,y) = 5\cdot \pi \cdot ({x^2} + {y^2})$, the linear tilt-shift is set as:$\delta = 0.75 + 0.05\cdot x + 0.15\cdot y$, the nonlinear tilt-shift is set as:$\delta = 0.75 + 0.05\cdot {x^2} + 0.15\cdot y$,where $\textrm{ - }1 \le x \le 1$, $- 1 \le y \le 1$. The phase distribution (RMS = 6.649 rad), the linear tilt-shift distribution (RMS = 0.0915 rad), and the nonlinear tilt-shift distribution (RMS = 0.0881 rad) are shown in Figs. 2(a)–2(c). We perform the calculation using the simulated data from Eq. (10) to illustrate the fitting parameters. Through the calculation, for linear tilt-shift the results are: ${x_0} ={-} 1.1084,{y_0} = 1.0995,{a_x} = 1.4414,{a_y} = 0.5777,\theta = 0.7908$, for non-linear tilt-shift the results are:${x_0} = \textrm{ - }1.1090,{y_0} = 1.0911,{a_x} = 1.3511,{a_y} = 0.7660,\theta = 0.7933$. The oblique-ellipse can be transformed into standard-ellipse with these parameters. Then we can correctly extract phase distribution from standard ellipse.

 figure: Fig. 2.

Fig. 2. Simulated phase distribution and phase-shift distributions. (a) The simulated phase distribution, (b) the simulated linear tilt-shift distribution, (c) the simulated nonlinear tilt-shift distribution.

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The comparison between the two algorithms with linear tilt-shift is shown in Fig. 3. The phase distributions calculated by the proposed algorithm (RMS = 6.6500 rad) and Wielgus’s algorithm (RMS = 6.6587 rad) are shown in Figs. 3(a) and 3(c), respectively. We can see that both proposed algorithm and Wielgus’s algorithm can effectively extract the phase distribution by comparing Figs. 3(a) and 3(c) with Fig. 2(a). That is to say, the two algorithms are insensitive to linear tilt-shift. However, the phase error distributions calculated by the proposed algorithm (RMS = 0.0667 rad) and Wielgus’s algorithm (RMS = 0.1312 rad) are different as shown in Figs. 3(b) and 3(d). The phase error of Wielgus’s algorithm is nearly 2 times compared to that of the proposed algorithm since the phase-shift calculated by nonlinear error functional minimization is an estimated value in Wielgus’s algorithm. Therefore, the proposed algorithm has higher precision than Wielgus’s algorithm when there is linear tilt-shift.

 figure: Fig. 3.

Fig. 3. The comparison between two algorithms with linear tilt-shift. (a) The unwrapped phase calculated by the proposed algorithm, (b) the phase error distribution calculated by the proposed algorithm, (c) the unwrapped phase calculated by Wielgus’s algorithm, (d) the phase error distribution calculated by Wielgus’s algorithm.

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The comparison between the two algorithms with nonlinear tilt-shift is shown in Fig. 4. The phase distributions calculated by the proposed algorithm (RMS = 6.6552 rad) and Wielgus’s algorithm (RMS = 5.7024 rad) are shown in Figs. 4(a) and 4(c), respectively. By comparing Figs. 4(a) and 4(c) with Fig. 2(a), the proposed algorithm can effectively extract the phase distribution. However, Wielgus’s algorithm fails to extract the phase distribution due to the nonlinear tilt-shift caused by nonlinearity of phase shifter. The phase error distribution calculated by the proposed algorithm (RMS = 0.0579 rad) as shown in Fig. 4(b) is smaller compared with that calculated by Wielgus’s algorithm (RMS = 4.7024 rad) as shown in Fig. 4(d). It indicates that the proposed algorithm can effectively extract phase distribution compared to Wielgus’s algorithm when there is nonlinear tilt-shift. Moreover, the phase error distributions between linear and nonlinear tilt-shift are similar as shown in Figs. 3(b) and 4(b). It indicates that the proposed algorithm can balance linear and nonlinear tilt-shift since phase shift errors are corrected by Eqs. (10) and (11).

 figure: Fig. 4.

Fig. 4. The comparison between the two algorithms with nonlinear tilt-shift. (a) The unwrapped phase calculated by the proposed algorithm, (b) the phase error distribution calculated by the proposed algorithm, (c) the unwrapped phase calculated by Wielgus’s algorithm, (d) the phase error distribution calculated by Wielgus’s algorithm.

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3.2 The performance analysis with Noises

In order to analyze the influence of noises on the proposed algorithm, different levels of noises generated by the function $awgn$ in MATLAB are added to the linear tilt-shift simulation of Section 3.1. The RMS phase error and processing time with different signal to noise ratio (SNR) of noises ranging from 13 dB to 60 dB are shown in Table 1. By analyzing Table 1, for the two algorithms, the RMS phase error decreases as the SNR of noises increases, and the processing time is nearly unchanged with different noises. We plot the RMS phase error with noises as shown in Fig. 5(a). The RMS phase error of the two algorithms is nearly unchanged when the SNR of noises is greater or equal to 30 dB. It indicates that the two algorithms are insensitive to noises with SNR $\ge $ 30 dB. And under the same SNR of noise, the phase error of proposed algorithm is smaller than that of Wielgus’s algorithm in Fig. 5(a). That is to say, the proposed algorithm has better anti-noise performance. We plot the processing time with noises as shown in Fig. 5(b). The computational time of proposed algorithm is less than that of Wielgus’s algorithm since the proposed algorithm can directly obtain the tested phase without pre-filtering. Therefore, the proposed algorithm can complete phase extraction faster than Wielgus’s algorithm.

 figure: Fig. 5.

Fig. 5. The RMS phase error and processing time with the two algorithms. (a) RMS phase error with different levels of noise, (b) consuming time with different levels of noise.

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Table 1. The RMS phase errors and processing time with different levels of noise

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3.3 The performance analysis with Disturbances

To verify the influence of disturbances on the accuracy of the proposed algorithm, the background intensity and the modulation amplitude are set as:

$$\begin{array}{l} {a_1}({x,y} )= exp\cdot ({ - 0.02({{x^2} + {y^2}} )} ),\ {b_1}({x,y} )= exp\cdot ({ - 0.02({{x^2} + {y^2}} )} )\\ {a_2}({x,y} )= (1 + k)\cdot exp\cdot ({ - 0.02({{x^2} + {y^2}} )} ),\ {b_2}({x,y} )= (1 + k)\cdot exp\cdot ({ - 0.02({{x^2} + {y^2}} )} )\end{array}$$
where, $k$ is the levels of fluctuation in the second interferogram ranging from 0 to 6. The phase distribution is the same as the simulation of Section 3.1 with linear tilt-shift. Table 2 shows the extracted RMS phase errors with different fluctuations. The RMS phase errors curves of proposed algorithm and Wielgus’s algorithm are plotted in Figs. 6(a) and 6(b), the RMS phase errors of the two algorithms changes slowly as the levels of fluctuation ranging from 0 to 1. However, it increases greatly as the levels of fluctuation ranging from 1.1 to 6. By comparing Figs. 6(a) and 6(b), the accuracy of Wielgus’s algorithm is smaller than proposed algorithm since the error of intensity-based filtration of the tilt-shift component is unavoidable in Wielgus’s algorithm. When fluctuations are greater than 1, the RMS phase errors of Wielgus’s algorithm is over 0.3 rad since the error caused by the approximated tilt-shift value is not fully compensated. It indicates that Wielgus’s algorithm is sensitive to large disturbances ($k \ge 1$). For the proposed algorithm, it has high accuracy (the RMS phase error is less than 0.07 rad) when fluctuations ranging from 0 to 6 because the phase-shift error is compensated successfully by transforming oblique-ellipse into standard-ellipse. Therefore, it’s reasonable to conclude that the proposed algorithm can balance the small and large background intensity and modulation amplitude fluctuations.

 figure: Fig. 6.

Fig. 6. Performance comparison between the two algorithms (a) RMS phase error of different methods with different levels of fluctuation (b) RMS phase errors of proposed algorithm with different levels of fluctuation.

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Table 2. RMS phase errors with different levels of fluctuation using different methods

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3.4 The performance analysis with the number of fringes

Some PSAs, such as the normalized difference maps (ND) [20] and the difference map normalization and diamond diagonal vector normalization (DN&DDVN) [21], extract phase distribution effectively when the number of fringes in one interferograms is greater than 1. To verify the influence of different fringes on the proposed algorithm, the phase distribution is set as:$\phi = m\cdot \pi ({x^2} + {y^2})$, where $m$ ranging from 0.6 to 10 represents the number of the fringes in one interferogram. The background intensity and modulation amplitude are set as:

$$\begin{array}{l} {a_1}({x,y} )= exp\cdot ({ - 0.02({{x^2} + {y^2}} )} ),\ {b_1}({x,y} )= exp\cdot ({ - 0.02({{x^2} + {y^2}} )} )\\ {a_2}({x,y} )= 2\cdot exp\cdot ({ - 0.02({{x^2} + {y^2}} )} ),\ {b_2}({x,y} )= 2\cdot exp\cdot ({ - 0.02({{x^2} + {y^2}} )} )\end{array}$$
the tilt-shift is set as:$\delta = 0.75 + 0.05\cdot x + 0.15\cdot y$. The RMS phase errors with different fringes are presented in Table 3, we plot the RMS phase errors with the two algorithms as shown in Fig. 7. The RMS phase errors of the proposed algorithm is smaller compared to that of Wielgus’s algorithm. For Wielgus’s algorithm, the phase errors are stable when m is greater than 2. However, the RMS phase errors increases as the number of fringes decreases when m is smaller than 2. It indicates that Wielgus’s algorithm performs well when the number of fringes is greater than 2. For the proposed algorithm, the phase errors are stable when m ranges from 0.7 to 10. It indicates that the proposed algorithm performs well when m is greater than 0.7. Therefore, the proposed algorithm is more flexible compared to Wielgus’s algorithm. As shown in Table 3, the RMS errors of Wielgus’s algorithm is more than 0.6 rad and that of the proposed algorithm is less than 0.14 rad when the number of fringes is less than 1. It’s reasonable to conclude that the proposed algorithm is more suitable for the case where the number of interference fringes is less than 1.

 figure: Fig. 7.

Fig. 7. The RMS phase errors of the two algorithms with different fringes.

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Table 3. RMS phase errors of different methods with different number of fringes

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3.5 The performance analysis with highly non-uniformity

In addition, to verify the influence of highly non-uniformity on the proposed algorithm, the spatially non-uniformity of $a$ and $b$ is 10 times compared to that in Section 3.4. The background intensity and the modulation amplitude are set as:

$$\begin{array}{l} {a_1}({x,y} )= exp\cdot ({ - 0.2({{x^2} + {y^2}} )} ),\ {b_1}({x,y} )= exp\cdot ({ - 0.2({{x^2} + {y^2}} )} )\\ {a_2}({x,y} )= 2\cdot exp\cdot ({ - 0.2({{x^2} + {y^2}} )} ),\ {b_2}({x,y} )= 2\cdot exp\cdot ({ - 0.2({{x^2} + {y^2}} )} )\end{array}$$
The phase distribution is set as:$\phi (x,y) = 5\cdot \pi \cdot ({x^2} + {y^2})$, the linear tilt-shift is set as:$\delta = 0.75 + 0.05\cdot x + 0.15\cdot y$, the nonlinear tilt-shift is set as $\delta = 0.75 + 0.05\cdot {x^2} + 0.15\cdot y$ . The phase distribution and phase error distribution with linear tilt-shift are shown in the Figs. 8(a) and (b), respectively. The phase distribution and phase error distribution with non-linear tilt-shift are shown in the Figs. 8(c) and (d), respectively. From Fig. 8, we can see that the proposed algorithm can correctly extract the phase distribution in the cases of linear and nonlinear tilt-shift. The follow results are obtained through this simulation: 1) for linear tilt-shift, the RMS phase error is 0.1004 rad, and the computational time is 0.042 s; 2) for non-linear tilt-shift, the RMS phase error is 0.0996 rad, and the computational time is 0.045 s. The results indicating that the proposed algorithm can accurately extract phase distribution with less time consuming. It’s reasonable to conclude that the proposed algorithm performs well when a spatial shape is highly non-uniformity.

 figure: Fig. 8.

Fig. 8. The phase distribution and phase error distribution with highly non-uniformity. (a) the unwrapped phase with linear tilt-shift, (b) the phase error distribution with linear tilt-shift, (c) the unwrapped phase with non-linear tilt-shift, (d) the phase error distribution with non-linear tilt-shift.

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3.6 The performance analysis with complex fringes

Finally, for further verify the feasibility of the proposed algorithm to complex fringes, the initial phase is changed as:

$$\phi (x,y) = 5\cdot \pi \cdot (3\cdot {x^2} + {y^2}) + 4\cdot peaks(512)$$
where, we use the function $peaks$ in MATALB. The background intensity, modulation amplitude and phase-shift value are same as the simulation of Section 3.4. The phase distribution is asymmetrical as shown in Fig. 9(a). The phase distributions calculated by the proposed algorithm and Wielgus’s algorithm are shown in Figs. 9(b) and 9(c). The phase error distributions calculated by the proposed algorithm and Wielgus’s algorithm are shown in Figs. 9(d) and 9(e). We can see that the two algorithms can effectively extract phase distribution by comparing Figs. 9(b) and 9(c) with 9(a). It indicates that the two algorithms are valid for the complex fringes. According to this simulation, the calculated results are as follow: 1) for the proposed algorithm, the RMS phase error is 0.0669 rad, and the computational time is 0.058 s; 2) for Wielgus’s algorithm, the RMS phase error is 0.3298 rad, and the computational time is 0.975 s. The results indicate that the proposed algorithm has higher precision and less time consumption than Wielgus’s algorithm. Moreover, the extracted phase error distribution (RMS = 0.0669 rad) as shown in Fig. 9(d) is almost the same as the circular fringes (RMS = 0.0667 rad) as shown in Fig. 3(b). It indicates that the proposed algorithm is suitable for both simple and complex fringes.

 figure: Fig. 9.

Fig. 9. Simulated result of complex fringes. (a) The simulated phase distribution, (b) the unwrapped phase calculated by the proposed algorithm, (c) the unwrapped phase calculated by Wielgus’s algorithm, (d) the phase error distribution calculated by the proposed algorithm, (e) the phase error distribution calculated by Wielgus’s algorithm.

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In summary, the proposed algorithm has the excellent performance according to the above results. 1) It can efficiently extract phase distribution with linear and nonlinear tilt-shift; 2) it has higher precision and less time consuming compared to Wielgus’s algorithm with different noises; 3) the accuracy of phase extraction is irrelevant to the levels of fluctuation; 4) phase can be extracted correctly when the number of fringes is less than 1; 5) the proposed algorithm is rapid and precise for simple and complex fringes.

4. Demonstration with experimental data

Some experiments have been performed to confirm the effectiveness of the proposed algorithm. Firstly, the straight fringe interferograms with the number of fringes more than 1 are collected to demonstrate the phase retrieval. The size of interferograms with random phase value captured by complementary metal oxide semiconductor (CMOS) is 2456*2058. After cropping by the function $imcrop$ in MATLAB, the new size of interferograms is 600*600. Figure 10(a) is one of the two interferograms, which suffers from strong background noise. By comparing Figs. 10(b) and 10(c), the proposed algorithm extract phase distribution more accurately compared to Wielgus’s algorithm. Moreover, the processing time of proposed algorithm is 0.062 s and that of Wilegus’s algorithm is 1.894 s. It indicates that the proposed algorithm is faster than Wielgus’s algorithm. That is to say, the proposed algorithm is suitable for fast dynamic measurements.

 figure: Fig. 10.

Fig. 10. Experimental results of straight fringe interferograms with the number of fringes more than 1. (a) One of the phase shifted interferograms, b) the unwrapped phase calculated by the proposed algorithm, (c) the unwrapped phase calculated by Wielgus’s algorithm.

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In addition, the straight fringe interferograms with the number of fringes less than 1 are also collected. The size of interferograms is 300*300. The phase distribution calculated by the proposed algorithm is more precise than that calculated by Wielgus’s algorithm as shown in Fig. 11. Besides, the processing time of proposed algorithm is 0.019 s and that of Wilegus’s algorithm is 0.451 s. It indicates that the proposed algorithm can extract phase distribution rapidly compared to Wielgus’s algorithm.

 figure: Fig. 11.

Fig. 11. Experimental results of straight fringe interferograms with the number of fringes less than 1. (a) One of the phase shifted interferograms, b) the unwrapped phase calculated by the proposed algorithm, (c) the unwrapped phase calculated by Wielgus’s algorithm.

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In order to verify the feasibility of the proposed method to fringes of different shapes, the circular fringe interferograms are collected. The size of interferograms is 300*300. The phase distribution calculated by the proposed algorithm is more precise than that calculated by Wielgus’s algorithm are shown in Fig. 12. Moreover, the processing time of the proposed algorithm is 0.021 s and that of Wilegus’s algorithm is 0.212 s, which indicates that the proposed algorithm can extract phase distribution rapidly compared with Wielgus’s algorithm. Through above experiments, we verify that the proposed algorithm can quickly and correctly extract phase distribution from two interferograms with different fringes, noise, disturbance and tilt-shift.

 figure: Fig. 12.

Fig. 12. Experimental results of circular fringe interferograms. (a) One of the phase shifted interferograms, b) the unwrapped phase calculated by the proposed algorithm, (c) the unwrapped phase calculated by Wielgus’s algorithm.

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5. Conclusion

In conclusion, a phase retrieval method from two-frame tilt-shift based on Lissajous ellipse fitting and ellipse standardization is demonstrated. It can rapidly extract phase distribution without filtering out the background intensity and calculating the phase shift. Ellipse standardization can compensate phase shift errors with the elliptic coefficients obtained by ellipse fitting. Then, the tested phase is extracted correctly from the standard-ellipse. We have compared the proposed algorithm with Wielgus’s algorithm by simulations and experiments. The proposed algorithm can rapidly and accurately extract phase distribution in a case where noise, non-uniformity, fluctuation and tilt-shift exist. In addition, the tested phase is extracted correctly when the number of fringes is less than 1. Finally, it is also proved that the proposed algorithm is effective for simple and complex fringes. Considering the simulated and experimental results, it is reasonable to conclude that the proposed algorithm is a powerful tool for rapid and precise phase retrieval with tilt-shift.

Funding

National Natural Science Foundation of China (61575051); Science and Technology Planning Project of Shenzhen Municipality (JCYJ20180306171923592, JSGG20190819175801678).

Disclosures

The authors declare no conflicts of interest.

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Figures (12)
Fig. 1.
Fig. 1. Lissajous figures. (a) The oblique ellipse (with errors), (b) the standard ellipse (without errors).

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Fig. 2.
Fig. 2. Simulated phase distribution and phase-shift distributions. (a) The simulated phase distribution, (b) the simulated linear tilt-shift distribution, (c) the simulated nonlinear tilt-shift distribution.

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Fig. 3.
Fig. 3. The comparison between two algorithms with linear tilt-shift. (a) The unwrapped phase calculated by the proposed algorithm, (b) the phase error distribution calculated by the proposed algorithm, (c) the unwrapped phase calculated by Wielgus’s algorithm, (d) the phase error distribution calculated by Wielgus’s algorithm.

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Fig. 4.
Fig. 4. The comparison between the two algorithms with nonlinear tilt-shift. (a) The unwrapped phase calculated by the proposed algorithm, (b) the phase error distribution calculated by the proposed algorithm, (c) the unwrapped phase calculated by Wielgus’s algorithm, (d) the phase error distribution calculated by Wielgus’s algorithm.

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Fig. 5.
Fig. 5. The RMS phase error and processing time with the two algorithms. (a) RMS phase error with different levels of noise, (b) consuming time with different levels of noise.

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Fig. 6.
Fig. 6. Performance comparison between the two algorithms (a) RMS phase error of different methods with different levels of fluctuation (b) RMS phase errors of proposed algorithm with different levels of fluctuation.

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Fig. 7.
Fig. 7. The RMS phase errors of the two algorithms with different fringes.

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Fig. 8.
Fig. 8. The phase distribution and phase error distribution with highly non-uniformity. (a) the unwrapped phase with linear tilt-shift, (b) the phase error distribution with linear tilt-shift, (c) the unwrapped phase with non-linear tilt-shift, (d) the phase error distribution with non-linear tilt-shift.

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Fig. 9.
Fig. 9. Simulated result of complex fringes. (a) The simulated phase distribution, (b) the unwrapped phase calculated by the proposed algorithm, (c) the unwrapped phase calculated by Wielgus’s algorithm, (d) the phase error distribution calculated by the proposed algorithm, (e) the phase error distribution calculated by Wielgus’s algorithm.

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Fig. 10.
Fig. 10. Experimental results of straight fringe interferograms with the number of fringes more than 1. (a) One of the phase shifted interferograms, b) the unwrapped phase calculated by the proposed algorithm, (c) the unwrapped phase calculated by Wielgus’s algorithm.

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Fig. 11.
Fig. 11. Experimental results of straight fringe interferograms with the number of fringes less than 1. (a) One of the phase shifted interferograms, b) the unwrapped phase calculated by the proposed algorithm, (c) the unwrapped phase calculated by Wielgus’s algorithm.

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Fig. 12.
Fig. 12. Experimental results of circular fringe interferograms. (a) One of the phase shifted interferograms, b) the unwrapped phase calculated by the proposed algorithm, (c) the unwrapped phase calculated by Wielgus’s algorithm.

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Tables (3)

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Table 1. The RMS phase errors and processing time with different levels of noise

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Table 2. RMS phase errors with different levels of fluctuation using different methods

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Table 3. RMS phase errors of different methods with different number of fringes

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Equations (17)

Equations on this page are rendered with MathJax. Learn more.

I 1 ( x , y ) = a 1 ( x , y ) + b 1 ( x , y ) ⋅ cos ⁡ [ ϕ ( x , y ) ]
I 2 ( x , y ) = a 2 ( x , y ) + b 2 ( x , y ) ⋅ cos ⁡ [ ϕ ( x , y ) + δ ( x , y ) ]
I 1 = a 1 + b 1 ⋅ cos ⁡ ϕ
I 2 = a 2 + b 2 ⋅ cos ⁡ ( ϕ + δ ) = a 2 + b 2 ⋅ cos ⁡ ϕ ⋅ cos ⁡ δ − b 2 ⋅ sin ⁡ ϕ ⋅ sin ⁡ δ
cos ⁡ ϕ = ( I 1 − a 1 ) / b 1
sin ⁡ ϕ = a 2 b 2 ⋅ sin ⁡ δ − a 1 ⋅ cos ⁡ δ b 1 ⋅ sin ⁡ δ + cos ⁡ δ b 1 ⋅ sin ⁡ δ ⋅ I 1 − 1 b 2 ⋅ sin ⁡ δ ⋅ I 2
1 b 1 ⋅ sin 2 δ ⋅ I 1 2 − 2 ⋅ cos ⁡ δ b 1 ⋅ b 2 ⋅ sin 2 δ ⋅ I 1 ⋅ I 2 + 1 b 2 2 ⋅ sin 2 δ ⋅ I 2 2  + ( 2 ⋅ a 2 ⋅ cos ⁡ δ b 1 ⋅ b 2 ⋅ sin 2 δ − 2 ⋅ a 1 ⋅ cos 2 δ b 1 2 ⋅ sin 2 δ − 2 ⋅ a 1 b 1 2 ) ⋅ I 1 + ( 2 ⋅ a 1 ⋅ cos ⁡ δ b 1 ⋅ b 2 ⋅ sin 2 δ − 2 ⋅ a 2 b 2 2 ⋅ sin 2 δ ) ⋅ I 2 + ( a 2 b 2 ⋅ sin ⁡ δ − a 1 ⋅ cos ⁡ δ b 1 ⋅ sin ⁡ δ ) 2 + a 1 2 b 1 2 − 1 = 0
a ⋅ I 1 2 + b ⋅ I 1 ⋅ I 2 + c ⋅ I 2 2 + d ⋅ I 1 + f ⋅ I 2 + g = 0
( ( I 1 − x 0 ) ⋅ cos ⁡ θ + ( I 2 − y 0 ) ⋅ sin ⁡ θ ) 2 a x 2 + ( ( I 2 − y 0 ) ⋅ cos ⁡ θ − ( I 1 − x 0 ) ⋅ sin ⁡ θ ) 2 a y 2 − 1 = 0
x 0 = b ⋅ f − 2 c ⋅ d 4 ⋅ a ⋅ c − b 2 , y 0 = b ⋅ d − 2 a ⋅ f 4 ⋅ a ⋅ c − b 2 a x = 2 ⋅ a ⋅ f 2 + c ⋅ d 2 + g ⋅ b 2 − b ⋅ d ⋅ f − 4 ⋅ a ⋅ c ⋅ g ( b 2 − 4 ⋅ a ⋅ c ) ( ( a − c ) 2 + b 2 − ( a + c ) ) a y = 2 ⋅ a ⋅ f 2 + c ⋅ d 2 + g ⋅ b 2 − b ⋅ d ⋅ f − 4 ⋅ a ⋅ c ⋅ g ( b 2 − 4 ⋅ a ⋅ c ) ( − ( a − c ) 2 + b 2 − ( a + c ) ) θ = { 0 f o r b = 0 a n d a < c π 2 f o r b = 0 a n d a > c 1 2 ⋅ arctan ⁡ ( b a − c ) f o r b ≠ 0 a n d a < c π 2 + 1 2 ⋅ arctan ⁡ ( b a − c ) f o r b ≠ 0 a n d a > c
{ sin ⁡ ϕ = X a x cos ⁡ ϕ = Y a y
ϕ = arctan ⁡ ( r ⋅ X Y )  =  arctan ⁡ ( r ⋅ ( I 1 − x 0 ) ⋅ cos ⁡ θ + ( I 2 − y 0 ) ⋅ sin ⁡ θ ( I 2 − y 0 ) ⋅ cos ⁡ θ − ( I 1 − x 0 ) ⋅ sin ⁡ θ )
a 1 ( x , y ) = 1 + 0.1 ⋅ e x p ( − 0.02 ( x 2 + y 2 ) ) ,   a 2 ( x , y ) = 0.9 + 0.21 ⋅ e x p ( − 0.02 ( x 2 + y 2 ) ) b 1 ( x , y ) = 1 + 0.1 ⋅ e x p ( − 0.02 ( x 2 + y 2 ) ) ,   b 2 ( x , y ) = 0.9 + 0.21 ⋅ e x p ( − 0.02 ( x 2 + y 2 ) )
a 1 ( x , y ) = e x p ⋅ ( − 0.02 ( x 2 + y 2 ) ) ,   b 1 ( x , y ) = e x p ⋅ ( − 0.02 ( x 2 + y 2 ) ) a 2 ( x , y ) = ( 1 + k ) ⋅ e x p ⋅ ( − 0.02 ( x 2 + y 2 ) ) ,   b 2 ( x , y ) = ( 1 + k ) ⋅ e x p ⋅ ( − 0.02 ( x 2 + y 2 ) )
a 1 ( x , y ) = e x p ⋅ ( − 0.02 ( x 2 + y 2 ) ) ,   b 1 ( x , y ) = e x p ⋅ ( − 0.02 ( x 2 + y 2 ) ) a 2 ( x , y ) = 2 ⋅ e x p ⋅ ( − 0.02 ( x 2 + y 2 ) ) ,   b 2 ( x , y ) = 2 ⋅ e x p ⋅ ( − 0.02 ( x 2 + y 2 ) )
a 1 ( x , y ) = e x p ⋅ ( − 0.2 ( x 2 + y 2 ) ) ,   b 1 ( x , y ) = e x p ⋅ ( − 0.2 ( x 2 + y 2 ) ) a 2 ( x , y ) = 2 ⋅ e x p ⋅ ( − 0.2 ( x 2 + y 2 ) ) ,   b 2 ( x , y ) = 2 ⋅ e x p ⋅ ( − 0.2 ( x 2 + y 2 ) )
ϕ ( x , y ) = 5 ⋅ π ⋅ ( 3 ⋅ x 2 + y 2 ) + 4 ⋅ p e a k s ( 512 )
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