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Abstract
In this paper, we proposed a random phase retrieval approach based on two differential interferograms and VU factorization. Firstly, three interferograms with arbitrary phase shifts are subtracted, and then two differential interferograms are obtained. On this basis, the phase related to the matrix is solved via VU factorization and the measured phase is obtained. Compared with the commonly used high-accuracy phase-shifting algorithms, this method does not require long-time iteration and approximate conditions, so there is no requirement for the shape and number of fringes of the interferogram. More importantly, it is an extremely stable and high-accuracy phase-shifting algorithm, which has great advantages for practical applications. Experiments and simulation analysis show that the proposed method is a random phase retrieval approach with good anti-noise and strong applicability.
© 2021 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Digital holographic microscopy is a technology to obtain the phase information of sample through the principle of interference or diffraction. The phase-shifting interferometry (PSI) developed on this basis is widely used in the fields of quantitative phase imaging (QPI), precision mechanical component detection and flow field measurement because of its advantages of non-contact, full-field and high accuracy [1–3]. In the process of phase demodulation, high-accuracy demodulation of interference fringe pattern is the key step to ensure accurate phase acquisition. In the earliest multi-step phase-shifting algorithms, including least square algorithm (LSA) [4], fixed step multi-step phase-shifting algorithms [5–7], they are widely used because of their advantages of high-accuracy and high-speed. However, such algorithms need to know the phase shift in advance or require the phase shift to meet the equal step distribution, and the accuracy of the phase solution depends on the accuracy of the phase shift. The air disturbance, the calibration deviations of the phase-shifting device, and the instability of the laser frequency will lead to the deviation of the phase shift.
The method of determining the phase shift and extracting the phase from the phase-shifting interferogram with unknown phase shift does not need to calibrate the phase shift in advance. Directly determining the phase shift and extracting the measured phase from the collected phase-shifting interferogram sequence can reduce the impact of vibration and air flow on phase measurement. So far, many high-accuracy phase-shifting algorithms with unknown phase shift have been proposed, including the advanced iterative algorithm (AIA) [8], principal component analysis (PCA) method [9], and the advanced principal component analysis (APCA) method [10]. In addition, independent component analysis (ICA) method [11] and normalization orthogonal algorithm based on the orthogonal characteristics of fringes [12], and self-calibration algorithm based on linear correlation [13] are often used to calculate the phase. In recent years, many multi-step phase-shifting algorithms based on ellipse fitting are also proposed for implementation of phase extraction with high-accuracy [14,15]. However, in the above methods, the AIA approach is time-consuming because it needs continuous iteration to meet the convergence conditions. The other approaches, PCA and APCA algorithms exhibit fast computing time but the computation accuracy is affected by the number of fringes. To address the influence of fringe sparsity on the calculation accuracy, many phase-shifting algorithms independent of the number of fringes have been proposed. Mid-band spatial spectrum matching (MSSM) algorithm [16] and phase shift search (PSS) algorithm [17] can reduce the influence of the number of interference fringes on the calculation accuracy to a certain extent, but the MSSM algorithm needs filtering and has requirements for the phase shift distribution; PSS algorithm has few restrictions, but it needs a certain search, which is relatively time-consuming. The dual-channel simultaneous spatial and temporal polarization phase-shifting interferometry [18] can extract the phase quickly and accurately without being affected by the shape and number of fringes, but it needs to be realized by a dual-channel interference system, which will directly affect the actual application. In addition, for another kind of multi-step phase-shifting algorithm with unknown phase shift, the phase shift needs to be solved first, and then the phase is solved. Such methods mainly include spatial Fourier transform [19], inverse cosine algorithm [20,21], 2-norm algorithm [22], 1-norm algorithm [23], etc. These approaches also have certain requirements for phase shift distribution and fringe distribution, and are relatively time-consuming. For simplicity, the above-mentioned methods are classified and listed in Table S1 (see Supplement 1 for more details).
Briefly, the above-mentioned methods are always affected by the distribution of phase shift, fringe distribution and the number of fringes to a certain extent. The calculation accuracy is unstable and the scope of application is limited, which is not suitable for practical phase detection. For these cases, a phase retrieval approach based on differential interferogram and VU factorization is proposed in this study. This method does not need long-time iteration, and the calculation accuracy is not limited by the number and shape of fringes. The second part analyzes the principle of the method. And the feasibility of the method is verified via simulation and experimental data in the third part. Finally, the advantages of the method in stability, anti-noise, different fringe shapes and numbers are analyzed in the fourth part.
2. Principle analysis
In PSI, the intensity of three interferograms with arbitrary phase shift can be expressed as:
where $a(x,y)$ and $b(x,y)$ represent the background and intensity terms of the interferogram, ${\theta _1}$ and ${\theta _2}$ are the phase shift difference from the first interferogram, respectively. After subtracting the last two interferograms from the first interferogram and removing the background, two differential interferograms can be obtained, which can be expressed as:The VU factorization method is applied to solve matrices ${\textbf V}$ and ${\textbf U}$. The solution of ${\textbf V}$ can be solved by the following matrix solution method when ${\textbf U}$ is known:
The process of factorization is as follows. First, we randomly preset the value of phase shift $\theta$, then ${\textbf u}$ and ${\textbf v}$ have the initial value and the initial matrix ${{\textbf U}_{\textbf 0}}$ accordingly. Then ${{\textbf V}_{\textbf 0}}$ can be obtained by using Eq. (7) and the known matrix of differential interferograms. Using the obtained ${{\textbf V}_{\textbf 0}}$, the initial vector distributions are calculated according to Eq. (6) and updated to ${{\textbf V}_{\textbf 1}}$. Then, the ${{\textbf U}_{\textbf 1}}$ can be obtained by substituting ${{\textbf V}_{\textbf 1}}$ into the following formula:
The above process completes an iteration. Repeating the above iteration process until the accuracy of convergence is achieved, the final ${\textbf V}$ and ${\textbf U}$ values will be output. Finally, the measured phase will be calculated by Eq. (6). Where the calculation error for convergence can be defined as:
where n represents the n-th iteration, and $n = 1,2,3\ldots $. ${\varphi _{m,n}}$ and ${\varphi _{m,n - 1}}$ are the phase values obtained in the current and previous iteration in the iterative operation.3. Simulation and experimental verification
Firstly, we use a set of simulation interferograms with a sample height of 36.6 rad to verify the feasibility of the method, in which the size of the interferogram is set to 300×300 pixels, and the background and modulation terms are set to $a(x,y) = 80exp( - 0.05({(x - 0.01)^2} + {y^2})) + 40$ and $b(x,y) = 100exp( - 0.05({x^2} + {y^2}))$. The preset phase distribution is defined as $\varphi (x,y) = {x^2} + {y^2} + 2.5peaks({300} )$ and the preset phase shift values are 1.5 rad and 5 rad respectively. To make the simulation interferogram close to the reality, Gaussian white noise is added to the interferograms and the signal-to-noise ratio (SNR) is 35 dB. Three-frame phase- shifting interferograms and the preset phase are shown in Fig. 1(a-d). To compare the calculation accuracy, phase-shifting VU factorization (PSI-VU) [2], AIA and APCA approach are used to solve the measured phase at the same time. The phase distributions calculated by the proposed phase retrieval approach based on difference map using VU factorization (DM-VU), PSI-VU, APCA and AIA approach are shown in Fig. 1(e-h). To quantitatively analyze the accuracy of the above approaches, the phase deviation distributions are obtained by subtracting the calculated phase distributions using the above four approaches from the preset phase, as shown in Fig. 1(i-l). At the same time, the root mean square error (RMSE) values are calculated and the calculation times are compared in Table 1. The calculation results shown in Fig. 1 and Table 1 clearly demonstrate that the proposed method has obvious advantages in accuracy, and the calculation time is one order of magnitude faster than the AIA, which proves that the proposed method is a high-accuracy and fast iterative algorithm.
Fig. 1. Simulation results. (a) (b) (c) Three simulated phase-shifting interferograms; (d) Preset reference phase; The phase distributions calculated using the (e) DM-VU, (f) PSI-VU, (g) APCA and (h) AIA method; The phase deviation results calculated using the (i) DM-VU, (j) PSI-VU, (k) APCA and (l) AIA method.
Table 1. RMSE values and calculation times calculated using different methods (simulation)
To further verify the feasibility of the method, two groups of different types of experimental interferograms are applied. Firstly, a set of interferograms without secondary phase distortion is used to verify the feasibility of the method and the measured sample is polystyrene sphere. Figure 2(a) shows one of the experimental interferograms. As the secondary phase distortion is removed, the secondary interference fringes brought by the micro-objective are eliminated, and only the interference fringes of the sample are retained, so the fringe density is sparse. The region with sample in the interferogram is intercepted for calculation, and the size of the intercepted region is 168 × 468 pixels. The reference phase (REF) is obtained by calculating 200-frame intercepted interferograms using AIA approach, as shown in Fig. 2(b). The proposed method, PSI-VU, APCA and AIA approach are used to calculate three phase-shifting interferograms, the obtained phase distributions and the deviation distributions are shown in Figs. 2(c-f) and (g-j). The results of phase deviation distribution clearly indicate that the phase deviation of the proposed method is the smallest, followed by PSI-VU and AIA method. APCA algorithm has the largest calculation deviation because the fringes in interferogram are sparse and do not meet its calculation conditions. To quantitatively analyze the above results, the calculation accuracy and calculation time of each method are presented in Table 2. The results indicate that the proposed method has obvious advantages in terms of calculation accuracy and the calculation time.
Fig. 2. Experimental results (Polystyrene Spheres). (a) One of the experimental interferograms; (b) Reference phase (REF); The phase distributions calculated using the (c) DM-VU, (d) PSI-VU, (e) APCA and (f) AIA method; The phase deviation results calculated using the (g) DM-VU, (h) PSI-VU, (i) APCA and (j) AIA method.
Table 2. RMSE values and calculation times of different methods in calculating different types of interferograms (Experiment)
Then, to estimate the influence of experimental interferograms with different fringe types and numbers on the effectiveness of the algorithm, a group of interferograms with dense fringes is selected for calculation, and the imaged sample is a letter T. Three phase-shifting interferograms collected in the experiment are shown in Fig. 3(a-c). In the collected series of phase-shifting interferograms, 120-frame phase-shifting interferograms are calculated as the reference phase using the AIA approach, as shown in Fig. 3(d). Similarly, the proposed DM-VU, PSI-VU, APCA and AIA method are used to calculate the three interferograms respectively. The obtained phase distributions and the phase deviation distributions are shown in Fig. 3(e-h) and Fig. 3(i-l). The RMSE values and calculation times of above-mentioned methods are also shown in Table 2. The above series of results demonstrate that the proposed DM-VU approach has advantage in calculation accuracy, and the calculation time is one order of magnitude faster than the AIA. In a word, the proposed DM-VU approach can maintain high accuracy regardless of the fringe density and shape in the experiment.
Fig. 3. Experimental results (Letter T). (a) (b) (c) Three experimental phase-shifting interferograms; (d) Reference phase (REF); The phase distributions calculated using the (e) DM-VU, (f) PSI-VU, (g) APCA and (h) AIA method; The phase deviation results calculated by (i) DM-VU, (j) PSI-VU, (k) APCA and (l) AIA method.
4. Performance analysis of the method
Firstly, we discuss the effects of the number and shape of fringes on the proposed method and the other three approaches. We simulated three groups of interferograms with different fringe shapes, namely straight fringe, circular fringe and peaks fringe, in which the number of stripes changed from 0.5 to 8 with an interval of 0.5 stripes, and all interferograms are added with the same level of Gaussian white noise. The proposed method, PSI-VU, APCA and AIA methods are applied to calculate the above interferograms respectively, and the curve of RMSE value with the number of fringes is plotted in Fig. 4. In the straight fringe, when the number of fringes is greater than 1, with the increase of the number of fringes, the accuracy variation trends of the four approaches are basically the same. But when the number of fringes is less than 1, only the proposed DM-VU approach maintains high accuracy, and the accuracy of the other three approaches is low. For circular fringe and complex fringe, regardless of the number of fringes, the accuracy of the proposed approach remains stable and high, while the other three approaches fluctuate obviously with the change of the number of fringes. Moreover, all the above-mentioned results show that the accuracy of the proposed approach remains almost unchanged in the above three groups of interferograms with different fringe shapes.
Fig. 4. Variation of RMSE value of different approaches with the fringe number under different shape fringe patterns.
Secondly, the anti-noise performance of the four above-mentioned approaches is also analyzed. Keeping the same simulation conditions as Fig. 1 but the phase is set as $\varphi (x,y) = {{4\pi peaks({x,y} )} / {[{max\{{peaks(300)} \}- min\{{peaks(300)} \}} ]}}$, the SNR value of the noise is changed from 20 to 44 dB, with an interval of 2 dB. Similarly, four different approaches are used to calculate interferograms at different noise levels, and the results are shown in Fig. 5. The curve distribution shows that the accuracy of the above four methods increases with the SNR value of the noise level; however, the proposed DM-VU method still has obvious advantage in accuracy regardless of the noise level.
Finally, the stability of the proposed method is analyzed by calculating 1000 groups of randomly shaped interferograms in this part. We use Gaussian function to randomly transform 1000 different surfaces as the sample to be measured, randomly superimpose different background and modulation terms, and set the random phase shift distribution to obtain 1000 groups of randomly distributed interferograms. The variation range of phase is randomly from 3 to 60 rad, and the variation range of phase shift is randomly from 0 to 6.28 rad. Figure 6 shows the RMSE values between the reference phase and the calculated phases obtained by calculating the above-mentioned 1000 groups of interferograms using different methods. Quantitatively, the RMSE performance of different methods counted under 1000 groups of interferograms are showed in Table 3. Here, when the RMSE value greater than 0.2 rad, we define that as calculation error, greater than 0.08 rad and less than 0.2 rad as poor recovery accuracy, and less than 0.08 rad as high recovery accuracy. Further, a Box plot is used to describe the accuracy distribution of 1000 groups of data and the stability of the four above-mentioned approaches. The statistical distribution of RMSE values is shown in Fig. 7. The maximum, minimum and mean values of all data calculated by each method are marked in the figure, and the mean values of RMSE calculated using the four above-mentioned approaches are 0.0089, 0.0292, 0.0567 and 0.0293 rad, respectively. Obviously, the average RMSE value calculated by the proposed DV-VU method is the smallest. In addition, the accuracy distribution range of the middle 500 groups of data is marked with a Box, which indicates that half of the accuracy distribution of 1000 groups of data is located in this range. Therefore, the length of Box can describe the accuracy distribution range for most of the calculation results. The results indicate that the Box length of the proposed method is the shortest, which prove that the accuracy of the proposed method is the most stable compared with other methods. From the above analysis, it can be concluded that DM-VU approach is effective and universal.
Fig. 6. RMSE values between the reference phase and the calculated phases obtained by calculating 1000 groups of randomly shaped interferograms using different methods.
Fig. 7. The statistical distribution of RMSE values obtained by calculating 1000 groups of randomly shaped interferograms using different methods. Where the Box represents the accuracy distribution range of the middle 500 groups of data, and each black line in the Box represents the mean value of RMSE calculated using 1000 groups of interferograms.
Table 3. The RMSE performance of different methods counted under 1000 groups of randomly shaped interferograms (group)
5. Conclusion
A random phase retrieval approach based on difference map using VU factorization method (DM-VU) is proposed. First, three interferograms with arbitrary phase shifts are subtracted, and then two differential interferograms can be determined. On this basis, we can achieve accurate phase retrieval using the VU factorization method. The important property of the VU factorization is that the convergence of VU factorization is robust to initial default, and it does not need approximate conditions, which exhibits high calculation accuracy and fewer iterations. In addition, the differential operation can further eliminate the influence of background term on the calculation accuracy. Consequently, compared with the existing phase-shifting approaches, the proposed approach has no requirements about the fringes number, fringes shape and the phase shifts. Both the simulation analysis and experimental research show that the proposed DM-VU approach has obvious advantages in the accuracy and stability, and less calculation time compared with the commonly utilized iterative algorithms. Importantly, the proposed DM-VU approach will provide a useful solution for the practical applications in optical measurement and quantitative phase imaging.
Funding
National Natural Science Foundation of China (61805086).
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.
Supplemental document
See Supplement 1
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