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Abstract
To obtain higher phase accuracy with less computation time in phase-shifting interferometry, a random phase-shifting algorithm based on principal component analysis and least squares iteration (PCA&LSI) is proposed. The algorithm does not require pre-filtering, and only requires two-frame phase-shifted interferograms and less computation time to obtain a relatively accurate phase distribution. This method can still extract the phase with high precision when there are few fringes in the interferogram. Moreover, it eliminates the limitation that the PCA algorithm needs more than three frames of interferograms with uniform phase shift distribution to accurately extract the phase. Numerical simulations and experiments confirm that the method is suitable for complex situations with different fluctuations in background intensity and modulation amplitude. And it can still achieve accurate phase extraction compared with other methods under different noise conditions.
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1. Introduction
Phase-shifting interferometry (PSI) is a powerful tool for precision testing with its high precision and non-contact characteristics [1]. The high performance of PSI is guaranteed by the precision and efficiency of the phase-shifting algorithm (PSA), and an excellent PSA should have two advantages which are high accuracy and timesaving feature [2]. Generally speaking, the PSA requires at least three interferograms as input, and the amount of phase shift between frames needs to be known accurately. However, due to the inevitable air turbulence, mechanical vibration, electromagnetic disturbance and even the nonlinear factors of the phase shifter, it is difficult to obtain an accurate phase shift amount, resulting in inaccurate final phase extraction [3–5].
Over the past few decades, to expand the application range of PSI, many PSAs have been proposed to deal with random phase shifts [6–14]. In 2004, Wang et al. proposed an advanced iterative algorithm (AIA) based on the least squares iterative process, which solved the limitations of the iterative algorithm by separating the frame-by-frame iteration from the pixel-by-pixel iteration , which provides a robust phase output [6]. In 2011, Vargas et al. applied the idea of dimensionality reduction in multivariate statistical theory to PSI, and proposed a principal component analysis (PCA) algorithm, which took the phase-shifted interferogram sequence as the initial variables. By solving the principal components of the initial variables, the phase information to be measured is then obtained. [7]. In 2020, Escobar et al. proposed a random PSA based on the principle of matrix decomposition, which takes the phase-shifted interferogram sequence as the original variable and quickly extracts its phase information through matrix decomposition [8]. However, all the above algorithms require at least three frames of interferograms.
Using a smaller number of interferograms can greatly reduce the hardware and environmental requirements, so many scholars try to solve this problem with a single-frame algorithm [15–20]. However, phase reconstruction using only a single interferogram is difficult, especially for those containing closed fringes, and the over-underdetermined equation makes many single-frame methods extremely computationally inefficient [21]. In addition, symbol ambiguity during single-frame demodulation has also become one of the bottlenecks hindering the development of single-frame interference technology [22]. Considering that the two-frame algorithm has higher measurement accuracy than the single-frame algorithm, and it is more important that two-frame are the minimum number of interferograms to reconstruct the phase without local sign ambiguity. Therefore, the two-frame algorithm is considered a good compromise between single-frame and multi-frame algorithms [23].
Since the two-frame PSA was proposed, it has received lasting attention from scholars [24–30]. Especially in the past ten years, the research on two-frame algorithms has continued to heat up [31–40]. From 2011 to 2012, Vargas et al. successively proposed three two-frame random PSAs: the self-tuning method (ST), the regularized optical flow method (OF) and the Gram-Schmidt orthogonalization method (GS). Especially the GS method, because of its speed and accuracy, has become one of the most popular phase reconstruction methods in two-frame interferometry [31–33]. In 2012, Deng et al. proposed a method to calculate the unknown phase shift using extreme value of interference (EVI), and then obtain the phase distribution through the arctangent operation [34]. In 2017, Tian et al. proposed a local mask-based method to estimate the amount of phase shift to achieve phase demodulation [35]. Around 2019, Cheng et al. proposed a method to directly estimate the amount of inter-frame phase shift by solving a fourth-order polynomial equation, and subsequently developed a coefficient of variation minimization (CVM) method based on minimizing the modulation term of the interferogram [37,38]. From 2020 to 2022, successively proposed the statistical averaging method (SA) of the two-frame random PSA and the simplified method of calculating the phase by first-order norm normalization [39,40]. However, many of the two-frame PSAs mentioned above require pre-filtering before phase extraction [41–43]. The pre-filtering process not only cost more time, but also the complex pre-filtering parameter settings will inevitably affect the accuracy [44,45]. Methods without pre-filtering may require more than three frames of interferograms, and more acquired images means more time. Generally, without considering the complexity of the PSI system, the PSA with more interferograms has higher accuracy and slower speed, while the PSA with fewer interferograms has faster speed and lower accuracy. To balance computational accuracy and computational efficiency, it becomes necessary to explore two-frame random PSAs with fewer interferograms and without pre-filtering.
In this paper, a random two-frame PSA based on principal component analysis and least squares iterative technique (PCA&LSI) is proposed. An approximate phase-shifted interferograms sequence is first constructed to estimate the initial phase, and then a more accurate phase distribution is extracted by least squares iteration. The algorithm does not require pre-filtering, and only requires two-frame phase-shifted interferograms with less processing time to obtain an accurate phase distributions. Numerical simulations and/or comparative experiments demonstrate the stability and accuracy of the proposed PCA&LSI method under different conditions. In the following, the proposed method is introduced in detail.
2. Principle of PCA&LSI
2.1 Two-frame PCA to estimate the initial phase
The intensity expressions of two-frame phase-shifted interferograms are:
where $x$ and $y$ are spatial coordinates, and ${I_n}(x,y)$ is the $n$th phase-shifted interferogram. ${A_n}(x,y)$ and ${B_n}(x,y)$ respectively represent the background intensity and modulation amplitude, $\varphi (x,y)$ and $\delta _{n}$ are the measured phase and the phase shift. In the following, the spatial coordinates $(x,y)$ has been omitted for convenience. Equation (1) can be rewritten as:From Eq. (2), we can see that the intensity of the phase-shifted interferogram can be expressed as a linear combination of two signals, and the background intensity which can be understood as an offset constant that does not affect the principal component information. Considering that PCA requires at least three frames of interferograms to realize the phase demodulation process, we need to reconstruct the interferograms based on the two frames of existing interferogram. We noticed that the fluctuations of the background intensity and modulation amplitude usually do not vary much between two frames of phase-shifted interferogram, so it can be assumed that ${A_1} = {A_2} = A$, ${B_1} = {B_2} = B$, and through the difference between the two frames of interferogram, we can get:
2.2 Least squares suppression of initial phase error
For a set of data, if the functional model is known, the values of each parameter in the functional model can be solved by means of data fitting. By extension to each frame of the interferogram, the phase shift can be determined from a known phase distribution using a least squares algorithm (LSA), which can be easily incorporated into an iterative process. The initial phase estimated by the PCA algorithm constructed in Section 2.1 can be combined with LSA to suppress the initial phase error. Here we redefine the phase-shifted interferogram:
where the subscript $(n = 1,2)$ denotes the number of the phase-shifted interferogram, $(k = 1,2,\ldots,K)$ denotes the individual pixel in each interferogram, ${\varphi _k}$ is the phase of the $k$th pixel, and ${\delta _n}$ are the phase shift of the $n$th interferogram.After the initial phase ${\varphi _k}$ is estimated, the phase shift ${\delta _n}$ can be easily determined by using the least squares strategy. Ideally, the background intensity and modulation amplitude in a frame of interferogram are constant variables. In practical conditions, compared with the cosine change of the phase distribution, the background intensity and modulation amplitude are still relatively slow variables. Therefore, it is assumed here that there is no intra-frame variation in background intensity and modulation amplitude, that is, no inter-pixel variation, ${A_{n1}} = {A_{n2}} = \cdots = {A_{nK}}$ and ${B_{n1}} = {B_{n2}} = \cdots = {B_{nK}}$, and they are just the functions of frames. For the $n$th interferogram, there exists: $a_n = {A_n}$, $b_n = {B_n}\cos {\delta _n}$ and $c_n = - {B_n}\sin {\delta _n}$, so Eq. (10) can be simplified to:
The sum of the squares of the differences between the measured intensity and the theoretical intensity of the interferogram can be expressed as:2.3 Iterative strategy improves phase measurement accuracy
Based on the method principles of Section 2.1 and Section 2.2, to further improve the accuracy of the phase calculation, this paper finally proposes the PCA&LSI method by introducing an iterative strategy. The PCA method has the advantage of fast measurement and eliminates the uncontrollable pre-filtering process. The LSA method has the advantage of easily introducing an iterative strategy, thereby achieving high-precision measurement requirements. This section effectively combines these two advantages, and briefly summarizes the process of the proposed PCA&LSI method as follows:
1) Construct the interferogram sequence, apply the PCA strategy to estimate the principal component components, and obtain the initial phase distribution from the Eq. (9);
2) For the initial phase distribution obtained in step 1), use the LSA strategy to calculate the inter-frame phase shift $\delta$, the background intensity ${A_n}$ and the modulation amplitude ${B_n}$. The phase distribution is recalculated by Eq. (19);
3) Repeat step 2) with the updated phase distribution until the termination condition shown below is satisfied:
where $\xi$ is the predefined iterative convergence threshold, $10^{-5}$ is selected in this paper, and the superscript $i$ represents the number of iterations. The corresponding phase distribution under the termination condition is the phase distribution $\varphi _ k$ to be measured.The PCA&LSI algorithm is illustrated in Fig. 1, and the dashed line part represents the iterative process.
3. Simulation of PCA&LSI
To verify the effectiveness of the proposed PCA&LSI method, we conduct a series of numerical simulations and compare the proposed method with two other excellent two-frame random PSAs without pre-filtering: LEF and PCA&LEF. In the following, all computations in this paper are performed with a 2.90 GHz Intel Core i7-10700 CPU and we use the Matlab software for coding. According to Eq. (1), two-frame interferograms with a size of 512 $\times$ 512 pixels are generated. For more generality, the background intensity and the modulation amplitude of two-frame interferograms are set as non-uniform terms, ${A_n} = N_A\exp [ - 0.02({x^2} + {y^2})]$, ${B_n} = N_B\exp [ - 0.02({x^2} + {y^2})]$, where the $N_A$ of $1$st and $2$nd interferograms are set as 1.22 and 1.18, and the $N_B$ of $1$st and $2$nd interferograms are set as 1.08 and 1.12, respectively. The reference phase and the random phase shift are preset with $\varphi (x,y) = 4\pi ({x^2} + {y^2})$ and $\delta$ = 1.256 rad. In addition, the random noise with the signal-to-noise ratio (SNR) of 35dB is added to the two-frame interferogram. In Fig. 2, the two frames of the generated interferogram and the reference phase are shown, respectively.
Fig. 2. Simulated two-frame phase-shifted interferograms and the reference phase. (a) The first interferogram, (b) the second interferogram, and (c) the reference phase.
In the first simulation, we use the proposed PCA&LSI method to extract the phase from the simulated interferograms, and compare with the extracted phase of the LEF, PCA&LEF method. The extracted phases of the three methods are shown in Figs. 3(a)-(c), respectively. As is shown in Figs. 3(a)-(c), it can be noticed that there is obvious “jitter” in the phase plane extracted by the LEF method, and there is some “jitter” in the phase extracted by the PCA&LEF method. The phase extraction results of the PCA&LSI method is much smoother and the result is closer to the reference phase. Figures 3(d)-(f) shows the difference between the three methods and the reference phase, respectively. As is shown in Figs. 3(d)-(f), the difference between the three methods and the reference phase gradually becomes smaller, which further indicates that the proposed method has higher phase extraction accuracy.
Fig. 3. The phase distributions calculated by (a) LEF, (b) PCA&LEF, and (c) PCA&LSI; and phase error distributions calculated by (d) LEF, (e) PCA&LEF, and (f) PCA&LSI.
To quantitatively compare the performance of each algorithm, we calculated the root mean square (RMS) errors between the extracted phase and the reference phase of the three methods, and counted the calculation time of the three methods, which is shown in Table 1. From Table 1, it can see that the RMS errors of the phase distribution obtained by LEF, PCA&LEF and PCA&LSI methods are 0.3600 rad, 0.1631 rad and 0.0253 rad, respectively. The corresponding processing time are 0.4369 s, 0.5210 s, and 0.2695 s, respectively. Comparing the phase RMS errors of the three methods, the measurement accuracy of the PCA&LSI method is more than 10 times higher than that of the LEF method, and more than 5 times higher than that of the PCA&LEF method. In terms of processing time, the processing time of the PCA&LSI method is reduced by 40% compared with the LEF method, and by 50% compared with the PCA&LEF method. To further explore the efficiency of the PCA&LSI algorithm, we analyze the iterative cycle of the LSI algorithm. We noticed that the LSI algorithm converged almost after two or three iterations in all simulations. It shows that the proposed PCA&LSI is a high-precision and time-saving two-frame random PSA.
Table 1. Phase RMS errors and processing time of different methods.
In the second simulation, we examine the robustness of the three methods under varying background intensities and modulation amplitude fluctuations. In the simulation, we set the background intensity and modulation amplitude as: ${A_n} = N_A\exp [ - R_{f}({x^2} + {y^2})]$, ${B_n} = N_B\exp [ - R_{f}({x^2} + {y^2})]$, where $R_{f}$ = (0.01, 0.05, 0.1, 0.2) represent different fluctuation sizes, and other conditions are the same as in the first simulation. We obtained the error distributions of the extracted phases for different methods in four cases, as shown in Fig. 4. The error distributions of different methods for extracting phase in four cases are shown in Fig. 4. The first column Figs. 4(a1)-(a4) is the error distribution map of the LEF method under different $R_f$, the second column Figs. 4(b1)-(b4) is the error distribution map of the PCA&LEF method under different $R_f$, and the third column Figs. 4(c1)-(c4) are the error distribution diagrams of the PCA&LSI method under different $R_f$. Comparing each column, we can notice that the error of the three methods increases gradually as the volatility increases. Through the horizontal comparison of the three methods, the proposed PCA&LSI method always has the highest accuracy, followed by the PCA&LEF method, and the LEF method has the worst accuracy. It shows that the proposed PCA&LSI method has stronger robustness than the other two methods.
Fig. 4. Comparison of the phase errors of different methods under different background fluctuations and modulation amplitudes. (a1)-(c1) $R_f$ = 0.01: LEF, PCA&LEF, PCA&LSI phase errors; (a2)-(c2) $R_f$ = 0.05: LEF, PCA&LEF, PCA&LSI phase errors; (a3)-(c3) $R_f$ = 0.1: LEF, PCA&LEF, PCA&LSI phase errors; (a4)-(c4) $R_f$ = 0.2: LEF, PCA&LEF, PCA&LSI phase errors.
In the third simulation, the effect of different fringe numbers on the phase calculation accuracy of LEF, PCA&LEF and PCA&LSI methods is investigated. In this simulation, we set the reference phase as: $\varphi (x,y) = K_{f}\pi ({x^2} + {y^2})$ , where $K_f=1,2,\ldots,20$ represents the number of different fringes, and the other conditions are the same as in the first simulation. In Fig. 5, the abscissa represents different fringe numbers, the ordinate represents the phase root mean square error, and the three curves represent the root mean square error of the phase extracted by the three methods under different fringe numbers. Numerous studies have shown that in the case of a small number of fringes, the traditional two-frame algorithm generally encounters the problem of inability to demodulate because the approximation cannot be satisfied. There are also multiple approximation operations in the PCA algorithm, and the number of fringes in the interferogram will have a great impact on the PCA phase extraction accuracy, especially when the number of interferogram frames is small. According to the curve analysis in Fig. 5, as the number of fringes increases, the phase extraction accuracy of the three algorithms does not change much. Therefore, it can be concluded that both the PCA&LEF and PCA&LSI methods can effectively suppress the measurement error introduced by the approximation operation in the PCA algorithm. Comparing the PCA&LEF and PCA&LSI methods, it is obvious that the PCA&LSI method has better error correction ability and obtains higher phase extraction accuracy. The results of the third simulation show that the PCA&LSI method is not only insensitive to the fringe number of the interferogram, but also can more effectively suppress the phase error introduced by the approximation operation. The PCA&LSI method is a two-frame random PSA with higher measurement accuracy.
In the fourth simulation, the effect of random noise on the phase extraction accuracy of the PSAs was considered. The simulation uses the $awgn$ function in MATLAB to add random noise of different SNR to the two-frame interferograms, and the other conditions are the same as in the first simulation. We analyzed the effect of different SNR noises from 10 dB to 60 dB on the phase extraction of the three methods, and the results are shown in Fig. (6). Comparing the results of the three methods, the overall error of the LEF method is the largest, and the phase error hardly changes with the change of the SNR, which indicates that the main error of the LEF method in the simulation comes from the fluctuation of the background intensity and the modulation amplitude. Again, the LEF method is sensitive to fluctuations in background intensity and modulation amplitude. The phase errors of the PCA&LEF and PCA&LSI methods decrease with the increase of the SNR, but the PCA&LSI method can always obtain better results than the PCA&LEF method, indicating that the PCA&LSI method can better suppress noise. From Fig. 6, we can also notice that when the SNR exceeds 45dB, the phase extraction error of the proposed PCA&LSI method is almost negligible, indicating that the PCA&LSI method has stronger noise robustness.
Four numerical simulations show that the proposed algorithm is robust in different situations. The PCA&LSI algorithm can 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 PCA&LSI algorithm can be summarized as: 1) No time-consuming pre-filtering operation; 2) Less sensitive to fluctuating background intensity and modulation amplitude; 3) Effectively repair the approximation error introduced when the number of fringes is small; 4) Partially suppress the effect of measurement errors introduced by random noise. In short, the proposed method is a time-saving and robust two-frame random PSA.
4. Experiment of PCA&LSI
To further verify the performance of PCA&LSA, the three methods are compared and tested using experimental interferograms. Without loss of generality, two sets of interferogram sequences are selected in the experiment, one is the straight fringes interferogram and the other is the circular fringes interferogram. Two sets of experimental interferograms used in the experiments are shown in Figs. 7(a1),(a2) and Figs. 7(b1),(b2). To provide a reference phase for quantitative performance estimates, 19 frames of experimental phase-shifted interferograms were acquired for each of the two sets of interferogram sequences. For the two sets of interferogram sequences, use multi-frame AIA [6] to extract the phase from the 19 frames interferogram sequence, respectively. And then the obtained phases were unwrapped as the two sets of reference phases using a phase unwrapping algorithm [46], which are shown in Figs. 7(c) and 7(d), respectively.
Fig. 7. The experimental interferograms: (a1),(a2) two-frame interferograms with straight fringes, (b1),(b2) two-frame interferograms with circular fringes, (c),(d) two sets of reference phase.
Figures 8 and 9 show the measurement results of the experimental straight fringes interferogram and the experimental circular fringes interferogram under three different methods, respectively. Figures 8(a)-(c) and Figs. 9(a)-(c) are the extracted phase distributions, Figs. 8(d)-(j) and Figs. 9(d)-(j) are the difference map of the extracted phase and the reference phase. Comparing the results in Fig. 8, the three methods have successfully achieved phase extraction. This is because the straight fringes interferogram in Figs. 7(a1),(a2) is relatively uniform, and then we will carry out a quantitative analysis to better compare the accuracy of the three methods. As for the results in Fig. 9, due to the obvious fluctuations in the background intensity and modulation amplitude in the circular fringes interferogram in Figs. 7(b1),(b2), the phase extraction of the LEF method almost fails, and the results of PCA&LSI are relatively better than those of PCA&LEF. From Fig. 8 and Fig. 9, it can be concluded that the proposed PCA&LSI method can obtain stable and higher phase extraction results in different situations.
Fig. 8. Experimental comparison of the straight fringes. The phase extracted by (a) LEF, (b) PCA&LEF, (c) PCA&LSI; and the differences between the reference and phase extracted by (d) LEF, (e) PCA&LEF, (f) PCA&LSI.
Fig. 9. Experimental comparison of the circular fringes. The phase extracted by (a) LEF, (b) PCA&LEF, (c) PCA&LSI; and the differences between the reference and phase extracted by (d) LEF, (e) PCA&LEF, (f) PCA&LSI.
The RMS errors of the phases obtained by the LEF, PCA&LEF and PCA&LSI methods are shown in Table 2, respectively. For straight fringes, the phase RMS error values of the three methods LEF, PCA&LEF and PCA&LSI are 0.3095 rad, 0.2636 rad and 0.2487 rad, respectively. For the circular fringes, the phase RMS error values of the three methods LEF, PCA&LEF and PCA&LSI are 0.5727 rad, 0.4342 rad and 0.3862 rad, respectively. In the experiment, both the straight fringes and the circular fringes obtained similar results to those in the simulation. The LEF method had the worst phase extraction accuracy, followed by the PCA&LEF method, and the PCA&LSI method had higher phase extraction accuracy.
Table 2. Phase RMS errors and processing time of different methods.
The processing time of the three methods is also counted in Table 2. For straight fringes, the processing times of the three methods of LEF, PCA&LEF and PCA&LSI are 0.1360 s, 0.3026 s and 0.0737 s, respectively. For circular fringes, the processing times of the three methods LEF, PCA&LEF and PCA&LSI are 0.1907 s, 0.5594 s and 0.1240 s, respectively. The PCA&LSI method proposed in this paper still maintains higher computational efficiency. This shows that the proposed PCA&LSI method is a fast and robust two-frame random PSA under experimental conditions.
5. Conclusion
This paper presents a two-frame random PSA without pre-filtering based on principal component analysis and least squares technologies. First, an approximate interferometric sequence is constructed, and the initial phase is approximately estimated by the PCA method. The phase error is then suppressed by the least squares, introducing an iterative policy to obtain a more accurate phase distribution. The algorithm does not require pre-filtering, and only requires two-frame phase-shifted interferograms and less processing time to obtain an accurate phase distribution. Numerical simulation and experiments verify that the PCA&LSI method can accurately extract the phase in a variety of complex situations, such as fluctuations in background strength and modulation amplitude, under different background noise, and different stripes. The proposed PCA&LSI method has potential application value in fast and stable two-frame random PSI.
Funding
National Natural Science Foundation of China (61827825).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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