## Abstract

Phase measurement techniques using a single-shot carrier fringe pattern play an important role in optical science and technology and have been widely used for various applications. In this paper, we focus on the comparative study of two major fringe analysis techniques, the sampling moiré (SM) and the windowed Fourier transform (WFT). While SM converts a single-fringe pattern to multiple phase-shifted moiré fringe patterns to extract the phase information in the spatial domain, WFT obtains the phase information in the windowed Fourier domain; thus, the two methods look entirely different. We evaluate the phase extraction errors of SM and windowed Fourier ridges (WFRs) as a typical WFT method for both linear and nonlinear phases with/without noise against the reference Fourier transform (FT) technique. For the simulated fringe patterns with linear or nonlinear phase and different random noise level, all the methods have high phase extraction accuracies. For a real experiment with more complicated phase and discontinuities, SM and WFR, both local methods, yield quite similar results and outperform FT.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. INTRODUCTION

Phase measurement plays an increasingly important role in optical science and technology and has been used for many applications, such as (i) noncontact three-dimensional (3D) shape measurement by fringe projection profilometry (FPP) [1,2], (ii) out-of-plane deformation measurement by reflectometry [3], (iii) full-field in-plane displacement and strain measurement of various materials and structures by getting carrier fringe patterns [4,5], (iv) precision inspection of optical components by phase-shifting (PS) digital holography [6], (v) noncontact reflective-index distribution measurement in optical waveguides by PS laser microscope [7], (vi) visualization and automatic detection of defect distribution in atomic structures [8,9], and so on.

Therefore, fringe pattern analysis [10] is crucial in all these techniques, as their success depends on the quality of the extracted phase distribution. Several fringe analysis techniques have been proposed for accurate phase extraction. The widely used PS technique [11] in the time domain uses multiple fringe patterns and calculates the phase information pixel-by-pixel. It provides high resolution but is relatively more difficult for dynamic measurement, since it requires multiple fringe patterns. Hence, techniques based on a single carrier fringe pattern emerged, among which, Fourier transform (FT) [12–15] extracts the phase by manipulating the Fourier spectrum and is the most widely used technique in this category. As an extension, windowed Fourier transform (WFT) [16,17] manipulates the windowed Fourier spectrum, which is usually more effective than manipulating the Fourier spectrum. In addition, some researchers have proposed the phase analysis by use of wavelet transform (WT) [18–20]. The WT technique utilizes either the one-dimensional continuous wavelet transform (1D-CWT) [19] or the two-dimensional continuous wavelet transform (2D-CWT) [20] to extract the phase of a fringe pattern. The review of fringe pattern phase recovery using 1D-CWT and 2D-CWT has been reported in Ref. [21] in detail. Similarly, other unique single-shot fringe pattern analysis techniques using a 2D phase differencing operator [22] or a multiple signal classification [23] have also been reported. Besides, spatial carrier phase-shifting (SCPS) [24–32] adopts the PS concept and treats adjacent pixels as a phase-shifted sequence with the carrier frequency as the phase-shift value, from which the phase is extracted by a PS algorithm. Naturally, this method requires prior knowledge about the carrier frequency, which is usually varying spatially and unknown. The sampling moiré (SM) method [33] is another useful technique that also resembles the PS algorithm for phase extraction, but this time, several phase-shifted moiré fringe patterns are obtained by down-sampling the carrier fringe pattern and later interpolating the intensity.

From the literature, FT, WFT, WT, SCPS, and SM are all shown to be useful,
each with its own advantages. It is thus natural and necessary to have a
comparative study to have a more in-depth understanding of these methods.
There are several excellent works on the comparative study of different
phase analysis techniques for fringe patterns. For instance, Gdeisat
*et al.* reported a comprehensive comparison study
focused on the 1D-CWT both temporally and spatially against FT, PS, and
WFT [34]. With a similar approach,
Huang *et al.* studied the comparison of FT, WFT,
and WT for phase extraction from a single-fringe pattern in FPP in detail
[35]. Besides, Agarwal and Kemao
also theoretically demonstrated that WFT can be seen as a special case of
an SCPS algorithm when the window function is taken as a 1D rectangular
window [36]. However, to date, no
comparison results of phase measurement accuracy for SM and WFT have yet
been reported. This is the motivation for our study.

In this paper, both SM and WFT as local analysis methods are compared with each other, with the well-known 2D-FT as a reference technique. There are two algorithms based on WFT: windowed Fourier ridges (WFRs) and windowed Fourier filtering (WFF). WFR is more suitable for processing a carrier fringe pattern [16], and thus is discussed in this paper. Since SM processes a fringe pattern in a row-by-row manner, to ensure fairness in comparison, 1D-WFR is selected for comparison. Both linear and nonlinear phase distributions, as well as different noise conditions, are considered. As a result, 1D-SM and 1D-WFT produce almost identical results in both simulation and experimental examples, while they perform similarly to 2D-FT in simulation tests but outperform it in the real experiment.

The rest of the paper is organized as follows. Section 2 presents the principles of 2D-FT, 1D-SM, and 1D-WFR for phase extraction from a carrier fringe pattern. Section 3 shows the comparison environment and simulation results based on both linear and nonlinear phases under various noise conditions. Section 4 compares the computation time of three methods. Section 5 compares the measurement accuracy based on experimental profilometry data. Section 6 further discusses these algorithms regarding various performance issues. Finally, Section 7 concludes the paper.

## 2. PRINCIPLES OF FT, SM, AND WFR

In this section, the carrier fringe pattern is first mathematically represented, followed by the introduction of the principles of 2D-FT, 1D-SM, and 1D-WFR.

#### A. Single-Shot Carrier Fringe Pattern

A 2D fringe pattern with a spatial carrier, $f(x,y)$, can be written as

#### B. Principle of 2D-FT

Since 2D-FT is widely used for processing a 2D carrier fringe pattern, we use it as a reference technique in our comparison. Its fundamental principle is as mentioned in [12,13]. First, the fringe pattern in Eq. (1) is rewritten as

#### C. Principle of 1D-SM

In SM, several phase-shifted moiré fringe patterns are obtained from a single-fringe pattern by a process called “down-sampling” and “intensity-interpolation” [33,37]. In this process, a 1D fringe pattern $f(x)$ represented by Eq. (2) is down-sampled at a regular sampling pitch $T$ into $T$-step down-sampled images. These down-sampled images are then interpolated to obtain $T$-step phase-shifted moiré fringe patterns by using 1-order, cubic, or high-order (such as $B$-spline function) interpolation, to make the moiré fringe patterns have the same size as the carrier fringe pattern. The 1-order and 3-order interpolations have been often used in practical application with good performance, and thus both of them are involved in the comparison. For simplicity, they are denoted as 1D-SM (1-order) and 1D-SM (3-order), respectively. For 1D-SM (1-order) and 1D-SM (3-order), the pixels used for interpolation have intervals of $T$ and $3T$, respectively.

The $t$th phase-shifted moiré fringe pattern, ${f}_{m,t}(x)$, can be described as [37]

#### D. Principle of 1D-WFR

In contrast to FT, which uses the frequency spectrum of the whole image, WFR processes a fringe pattern window by window. Given a row of fringe pattern represented by Eq. (2), its 1D WFT for 1D is

where $u$ and $x$ are the spatial pixel coordinates, ${\xi}_{x}$ is the frequency, and $g(x)$ is a Gaussian window function defined asInterestingly, the local frequency, defined as ${\omega}_{x}=\frac{d\phi (u)}{du}={\omega}_{c}+\frac{d{\phi}_{0}(u)}{du}$, can be immediately estimated from the windowed Fourier spectrum as

Subsequently, the phase can be extracted as## 3. COMPARISON BY SIMULATION

In this section, with the help of simulated fringe patterns, we compare the performance of 2D-FT, 1D-SM (1-order/3-order), and 1D-WFR. The fringe generation and parameter selection of these algorithms are elaborated first so that the comparison is representative, fair, and reproducible. Next, the comparison of different phase distributions and noise levels will be carried out.

#### A. Fringe Generation and Parameter Selection

2D fringe patterns are simulated according to Eq. (1). The background and amplitude are constant, while the phase is simulated using the MATLAB “peaks” function together with a carrier:

where $M$ represents the image size of the fringe pattern to be $M\times M$ and $k$ indicates the amount of complexity. In our simulation, $M=256$ and $P=8.1\text{\hspace{0.17em}}\text{pixels}$ are considered as a typical example and thus are preset. When $k=0$, the phase is perfectly linear, and the fringes are straight, as shown in Fig. 2(a). When $k$ increases to 0.5, 1.0, and 2.0, the phase becomes nonlinear and more complicated, as shown in Figs. 2(b)–2(d). Such fringe patterns are commonly seen in real applications.The implementation details for the three methods are as follows:

- (i) 2D-FT is implemented according to Refs. [12,13]. The most important parameter is the bandwidth of the BPF in the frequency domain, which is manually decided to minimize the phase error. To make the results reproducible, we used a window size of $61\text{\hspace{0.17em}}\text{pixels}\times 61\text{\hspace{0.17em}}\text{pixels}$ around the carrier frequency.
- (iii) 1D-WFR is implemented according to Ref. [16]. Although the window size is recommended to be ${\sigma}_{x}=5\sim 10$, throughout this paper, ${\sigma}_{x}=T/2$ is used to make window size comparable to 1D-SM (3-order). The 1D-WFR algorithm searches the ridge location of the windowed Fourier spectrum by scanning local frequency $\xi $ from a low bound ${\xi}_{xl}$ to a high bound ${\xi}_{xh}$ with a scanning step of ${\xi}_{xi}$. The frequency band for ridge search can be set as the default value of $[{\xi}_{xl},{\xi}_{xh}]=[0,2]$, but it can also be set more narrowly to reduce the search time without sacrificing the accuracy. The frequency scanning step is set as the default value of ${\xi}_{xi}=0.025$, which is designed so that the phase error due to scanning step is restricted to 0.1% of $2\pi $. This scanning step can be set even more finely for higher accuracy if necessary. In this simulation, we set $[{\xi}_{xl},{\xi}_{xi},{\xi}_{xh}]=[0.685,0.025,0.885]$ in a scanning range of $2\pi /T\pm 0.1$.

After getting phase distributions with these algorithms, the phase error distributions are calculated first as follows:

#### B. Performance Evaluation on Noiseless Fringe Patterns

### 1. Linear Phase

We start by applying all the methods to the fringe pattern with a linear phase (i.e., $k=0$), as shown in Fig. 2(a). The RMS phase errors of 2D-FT, 1D-SM (1-order), 1D-SM (3-order), and 1D-WFR are $7.6\times {10}^{-3}$, $3.35\times {10}^{-5}$, $1.64\times {10}^{-9}$, and $9.57\times {10}^{-15}\text{\hspace{0.17em}}\mathrm{rad}$, respectively. The significant error in FT is because FT assumes that the image being processed repeats itself in the spatial domain, which generates discontinuities at image borders when the fringe number in the image is not an integer. If the number of fringes in the image happens to be an integer, e.g., $P=12.8$ and $M/P=20$, we have tested that the phase error is on the order of ${10}^{-14}$, which means that FT is ideal. Since 1D-SM and 1D-WFR are both considered as local processing methods, the influence of image borders is confined to the pixels near these borders. As the SM fringe patterns have a low spatial frequency, i.e., the intensity varies slowly, the 3-order interpolation suffices. The 1-order interpolation gives a relatively higher error, but it will show an advantage on random noise, as will be seen soon. To conclude, from this simulation result, all three methods, including 2D-FT, 1D-SM (1-order/3-order), and 1D-WFR are theoretically valid.

### 2. Nonlinear Phase

Figure 3 shows the simulation result of RMS phase error when $k$ is changed from 0.0 to 2.0 for the noiseless case. In Fig. 3, the gray zone indicates that the phase error is less than 1% of $2\pi $ for general measurement applications. In contrast, the light-pink zone indicates that the phase error is less than 0.1% of $2\pi $. Such an area is important for high-precision measurement, especially for a derivative of phase that is of interest for strain measurement. For 2D-FT, we observe that the phase error is almost invariant for simulated fringe patterns with different $k$, and the phase error is 0.0076 rad, which corresponds to 0.12% of $2\pi $. The phase error is mainly due to the image border, as mentioned earlier. For SM, the phase error of 1D-SM (1-order) increases linearly and that of 1D-SM (3-order) increases exponentially when $k$ becomes large. 1D-SM (3-order) is more suitable for a complicated fringe pattern, compared with 1D-SM (1-order). For 1D-WFR, the phase error is about $4.3\times {10}^{-3}\sim 4.9\times {10}^{-3}\text{\hspace{0.17em}}\mathrm{rad}$, which corresponds to $0.07\sim 0.08\%$ of $2\pi $. When 1D-SM (3-order) and 1D-WFR are compared, the accuracy of 1D-SM (3-order) is similar to that of 1D-WFR when $k\le 1.0$. Otherwise, if $k$ is larger than 1.0, 1D-WFR is better than 3-order SM, since WFR has a scanning function by searching the best frequency to be analyzed.

While Fig. 3 shows the overall performance of 2D-FT, 1D-SM (1-order/3-order), and 1D-WFR, Fig. 4 depicts their detailed results of the 64th row cross section of the nonlinear phase with $k=1.0$, corresponding to the yellow line in Fig. 2(c). For 2D-FT, a periodical phase error occurs caused by the discontinuity, since the fringe number in the image is not an integer. Besides, due to the mismatch of the grating pitch and the sampling pitch, a periodical phase error easily occurs in 1D-SM (1-order) [37]. However, the phase error of 1D-SM (3-order) is much smaller and smoother, which is very similar to the 1D-WFR result. Thus, for an ideal noiseless case, both 1D-SM (3-order) and 1D-WFR provide similarly high accuracy for both linear and nonlinear phases.

#### C. Accuracy for Fringe Patterns in the Presence of Noise

Noise is unavoidable in practical phase measurement. Generally, for a commercial low-cost CCD camera, two or three intensity gray levels will be changed for an 8-bit (256 gray-level) image, corresponding to 1% noise level. Besides, measurement of low-reflective object or unsuitable setting of the camera such as underexposure or extremely high ISO would introduce a higher noise level. Accordingly, the 2D captured fringe pattern can be represented as

where $n(x,y)$ is the additive random noise. We assume the noise has a mean of zero and a standard deviation ranging from 0% (noiseless) to 40%, to compare the sensitivity of all the concerned algorithms towards the noise in more extensive applications.### 1. Linear Phase

For a fringe pattern with linear phase as $k=0$ in Eq. (14), the RMS phase error with the presence of different levels of noise is depicted in Fig. 5(a). The gray and light-pink zones indicate that the phase error is less than 1% and 0.1% of $2\pi $, respectively. This result is averaged over 10 cases for the same amount of noise. As shown in the gray zone in Fig. 5(a), all methods can determine the phase within 1% RMS phase error if the random noise is less than 15%, indicating their effectiveness in phase analysis.

For 2D-FT, the noise spreads across the entire spectral domain, making it difficult to accurately filter the fundamental component. It is interestingly noticed that FT outperforms all other methods in this comparison, mainly because the 2D algorithm using the whole image intensity information is more effective in noise suppression than the 1D algorithms using only local fringe information. For a heavy noise, 1D-SM (1-order) is slightly better than 1D-SM (3-order) because the 1-order interpolation is directly interpolated linearly, whereas high-order interpolation will introduce some interpolation error due to random noise.

### 2. Nonlinear Phase

The above noise testing is repeated for fringes with complex peaks phases. The particular example with $k=1.0$ is shown in Fig. 5(b). In general, first, 1-D SM (1-order/3-order) and 1D-WFR show almost identical results and trends, and second, the results are very similar to those of the linear phase in Fig. 5(a), showing that the noise is more impactful than the examined level of the complexity of a nonlinear phase.

As noticed in Fig. 5, 2D-FT offers better performance than 1D-SM and 1D-WFR. It should be noted that 2D-WFR is more tolerant of noise than the 2D-FT algorithm since 2D window uses more intensity data, as already reported in the early study that compared the measurement accuracy of FT and WFT (see Fig. 3 in Ref. [35]).

To further test the impact of noise, an example with a peaks phase for a noise $\text{level}=1\%$ (to be considered in general experimental environment) is shown in Fig. 6. As shown in Fig. 6(a), 2D-FT has some periodical phase error in the horizontal direction. It can be observed that SM (3-order) is more effective than 1D-SM (1-order) in extracting a nonlinear phase, which is quite smooth and similar to that of 1D-WFR.

Finally, we have to notice that, to pursue the accuracy of 0.1% of $2\pi $ (in the light-pink zone) for the nonlinear phase, not only the level of noise but also the complexity of phase needs to be restricted, which exposes us a research challenge in real measurement for future exploration.

## 4. COMPARISON OF COMPUTATION TIME

The computational efficiency is also an essential aspect of fringe pattern analysis. Here, we investigated and compared three methods using the same image and the same computer environment in MATLAB.

Table 1 indicates the comparison of computation time using the simulated fringe patterns [Figs. 3 and 5(b)] for three methods including 2D-FT, 1D-SM, and 1D-WFR. The calculation time is the average of 10 runs. We used a Windows 7 PC (64-bit; Xeon CPU E5-1650, 3.2GHz) and MATLAB code (Ver. R2014b) to test the computation time. As a result, 2D-FT as a global method is the fastest among the three methods. For 1D-SM, we confirmed that 1D-SM (3-order) is 1.5 times slower than 1D-SM (1-order), since a high-order interpolation needs to use more data and curve fitting. Compared with 1D-SM, 1D-WFR take more time, since it requires a 1D convolution and scanning process to estimate the local frequency.

## 5. COMPARISON BY EXPERIMENT

The comparison is extended to a real fringe pattern from a FPP system [39–43]. Experimental setup and analysis condition are introduced first, and then the comparison results are given.

#### A. Setup and Analysis Conditions

The FPP system we used includes a liquid crystal display (LCD) projector (EPSON, EB-1761W, 3LCD, WXGA $1280\text{\hspace{0.17em}}\text{pixels}\times 800\text{\hspace{0.17em}}\text{pixels}$, 2600 lm) and a complementary metal oxide semiconductor camera (Imaging Source Co., DFK 72BUC02, $1600\text{\hspace{0.17em}}\text{pixels}\times 1200\text{\hspace{0.17em}}\text{pixels}$ at 13 fps). The camera is mounted with an 8 mm focal-length megapixels lens (Computar, M0814-MP). The camera and the projector are situated at 800 mm from the object with an angle of about 30 deg between the projector and camera. An example of the experimental fringe pattern is shown in Fig. 7.

To have a benchmark, the $N$-step phase-shifting measurement (PSM), which has been proven to have high reliability and high resolution, is used. The phase-shifted fringe patterns are mathematically represented as

Some other experimental analysis conditions are as follows:

- (1) To compare the phase analysis results without shadow areas, we focused on a block of fringe pattern with a size of $400\times 800$ pixels, as indicated by the green rectangle in Fig. 7.
- (2) For 1D-SM (1-order/3-order), a down-sampling pitch $T=15$ is considered an optimal parameter, since the projected fringe pattern is around 15 pixels.
- (3) For the 1D-WFR to work effectively, 2D-FT is performed before 1D-WFR. The role of 2D-FT is to suppress the negative frequency and bias component, while the role of 1D-WFR is to retrieve the phase. The former helps the latter to determine the ridge of the fundamental frequency component accurately.

#### B. Comparison Study for Experimental Results

All the methods including 2D-FT, 1D-SM, and 1D-WFR are then applied to the experimental carrier fringe pattern, and the results are shown in Fig. 9. Note that unlike the eight-step PSM, which uses all eight phase-shifted fringe patterns, only the first fringe pattern is used for other methods. Figure 9(a) shows the Fourier spectrum by 2D-FT, and the first frequency component is extracted to calculate the phase distribution; Figures 9(b) and 9(c) show the moiré fringe by 1-order and 3-order intensity interpolation, respectively. We can see that the moiré fringe obtained by 3-order interpolation is smoother than that of 1-order interpolation. Figure 9(d) shows the ridge map obtained by 1D-WFR to find the optimal local frequency; Figs. 9(e)–9(h) show the phase distributions by 2D-FT, 1D-SM (1-order), 1D-SM (3-order), and 1D-WFR, respectively.

The phase gradients for 2D-FT, 1D-SM (1-order/3-order), and 1D-WFR are calculated and shown in Figs. 10(a)–10(d), respectively. As shown in Fig. 10(a), 2D-FT results in fluctuations on the object surface and hence leads to some periodic measurement error due to the complicated surface profile and discontinuities. As observed from Figs. 10(b)–10(d), 1D-SM (1-order/3-order) and 1D-WFR methods, on the other hand, provide similar results around the smooth face region.

For quantitative evaluation, the phase difference between these methods and the eight-step PSM is, respectively, calculated and shown in Figs. 10(e)–10(h). The yellow area indicates the absolute phase difference exceeds 5%. For the smooth area of the face, all methods are again shown to be satisfactory. However, for the areas at the nose part and edge of the face where discontinuities exist, all single-shot methods, 2D-FT, 1D-SM (1-order/3-order), and 1D-WFR, perform much worse than the eight-step PSM, showing the clear limitation of these methods.

To further examine the phase difference of the smooth part of the phase, the phase gradients of eight-step PSM, 1D-SM (3-order), and 1D-WFR at the cross section ${\mathrm{AA}}^{\prime}$ [as shown in Figs. 8(d) and 10(c)] are plotted in Fig. 11. The almost identical results from 1D-SM (3-order) and 1D-WFR are immediately observed and are better than that of eight-step PSM. The large standard deviation of eight-step PSM is due to the relatively small number of points (eight points from eight fringe patterns in this case) in its pixel-by-pixel phase computation manner. The almost identical results between 1D-SM (3-order) and 1D-WFR that have been observed in simulation are revealed again in experimental data, which is the most important and exciting finding in this study. These results can be attributed to the fact that both methods have a reasonable and effective phase analysis principle and their local processing manner.

## 6. DISCUSSION

The comparisons based on both simulated and experimental fringe patterns are summarized as follows:

- (i)
**Theoretical accuracy**: All the methods, 2D-FT, 1D-SM, and 1D-WFR have high theoretical accuracy. - (ii)
**Performance on complicated phase**: 2D-FT has unsatisfactory performance on complicated phase distribution and discontinuities. Although 1D-SM and 1D-WFR do not perform well in such a scenario, the local processing nature can isolate bad regions not to propagate to other parts. - (iii)
**Performance on noise**: All the methods present outstanding noise suppression capability. - (iv)
**Novel finding**: Most interestingly, the almost identical performance of 1D-SM and 1D-WFR is observed. - (v)
**Applications**: 1D-SM has found many applications, such as accurate in-plane displacement and strain of various materials [5,44–46] and structural health monitoring of large-scale structures [47], or removal of vibration in PS digital holography [48] with fast speed by use of an optimal sampling pitch and the exact physical grating size as prior knowledge. Similarly, WFR also has various successful applications [49]. Based on the findings from this study, both SM and WFR are encouraged for trial.

Besides, the following points are interesting to mention, although not emphasized in this paper:

- (i)
**Improvement of performance**: WFR can process more general fringe patterns. By incorporating the scanning of sampling pitch, SM is expected to handle more complicated fringe patterns in the future work. - (ii)
**Improvement of speed**: On the contrary, SM has demonstrated a lot of useful practical applications and showed very high accuracy and speed when the grid pitch is known in advance. If such prior knowledge is given to WFR, then a narrower frequency band for ridge searching can be set, resulting in similar high accuracy and speed. - (iii)
**Extension to the 2D analysis**: Based on the measurement accuracy of their 1D counterpart, 2D-SM [50] and 2D-WFR [16] are expected to have similar measurement accuracy to each other in their 2D version. The 2D algorithms may work worse when the phase distribution along the $y$ axis is complicated, but they are generally more robust to noise and useful to analyze speckle fringe pattern in electronic speckle pattern interferometry (EPSI) or digital holography.

## 7. CONCLUSIONS

Among several single-shot fringe pattern analysis methods for various applications, the phase measurement accuracy of two widely used techniques, namely, 1D-SM and 1D-WFR, are first evaluated and compared with the reference 2D-FT technique in this paper. First, we compared the theoretical accuracies without noise for linear and nonlinear phases by simulation. Then, the effect of noise is considered in the comparison, since noise is inevitably encountered in practice. A real experimental fringe pattern is also processed for comparison in terms of phase and phase gradient evaluation. We confirm that the phase measurement accuracies of 1D-SM and 1D-WFR are quite similar, despite their different ideas and algorithms, i.e., down-sampling and intensity interpolation for SM, and window-based scanning and selection of peak frequency for WFR. Consequently, users are encouraged to try both 1D-SM and 1D-WFR in their practice, as well as their 2D versions.

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