## Abstract

In this study, a novel, fast, and accurate in-plane displacement distribution measurement method is proposed that uses a digital camera and arbitrary repeated patterns based on the moiré methodology. The key aspect of this method is the use of phase information of both the fundamental frequency and the high-order frequency components of the moiré fringe before and after deformations. Compared with conventional displacement methods and sensors, the main advantages of the method developed herein are its high resolution, accuracy, speed, low cost, and easy implementation. The effectiveness is confirmed by a simple in-plane displacement measurement experiment, and the experimental results indicate that an accuracy of 1/1000 of the pitch can be achieved for various repeated patterns. This method is useful for various applications ranging from the study of displacement and strain distributions in materials science, the biomimetics field, and mechanical material testing, to secure the integrity of infrastructures.

© 2014 Optical Society of America

## 1. Introduction

The displacement distribution of materials and structures provides useful information in understanding the deformation behavior in materials science at the nano-scale [1] and in verifying the integrity of infrastructures at the macroscopic scale [2]. Moiré method is one effective and potential full-field optical technique for in-plane deformation measurement and strain analysis in experimental mechanics for a long time. Burch and Forno utilized a high-sensitivity moiré grid technique [3] to study deformations in large objects. Avril *et al*. presented an application of the grid method [4] to the assessment of crack initiation and growth in brittle materials.

To observe the finer moiré fringe, Post *et al*. proposed a moiré interferometry technique [5], and the thermal deformation of electronic interconnections and packages was successfully measured using this technique [6]. Kishimoto *et al.* developed an electron moiré method [7] to evaluate the microscopic mechanical property of materials by using a scanning electron microscope (SEM). Xie *et al.* [8] developed a method to observe the moiré fringe using a scanning confocal laser microscope (SCLM). To fabricate nano/micro model grids, a number of researchers have proposed several useful techniques, such as electron beam lithography [9], nanoimprint lithography [10, 11], and femtosecond laser exposure to metals and metal-based composite materials [12].

Recently, a simple and accurate phase measurement technique called the sampling moiré method [13] was developed for small displacement distribution measurements using cosinusoidal (gray) or rectangular (black-and-white) grating patterns. In this method, the measurement accuracy achieved 1/1000 of the grating pitch. A number of practical applications of the sampling moiré method have been successfully demonstrated through the measurement of the shape and strain of rotating tires [14], small deflections of a 10-m-long crane [15], and the thermal deformation of high-temperature piping in thermal power plants [16]. However, it is not easy to fabricate or attach fine gratings to nano-scale materials or mega-scale infrastructures. In contrast, various periodic patterns in materials and structures can be seen in our daily lives. For instance, the atomic structure of metals can be observed using high-magnification field emission scanning electron microscopy (FE-SEM) at the nano-scale; the periodic structures on the surface of butterfly wings [8] or the eye structure of a number of insects can be seen under microscopy as well as tile walls or windows of building at the macroscopic scale.

Wang *et al.* developed a spot moiré fringe [17] to determine the domain boundaries and structural parameters in ordered nanoporous structures. The pitches and the orientations of the nanopore arrays in three directions can be simultaneously determined in a large field of view by observing the repeated pattern onto the specimen itself. Recently, Patorski *et al.* discussed the phenomenon of moiré fringe phase multiplication for quasi-periodic structures based on the spatial beating effect between the *M*-th harmonic of the nonsinusoidal profile of a quasi-periodic specimen and the first harmonic of the reference grating [18]. In their study, both the digital moiré (with a computer generated reference grating) and the sampling moiré techniques were used for atomic force microscope (AFM) investigations of intermediate polymer stamps used in nanoimprint lithography. In addition, a mathematical model for an arbitrary quasi-periodic pattern and a Fourier series approach was suggested to represent the moiré phenomenon with higher harmonics.

To broaden the aforementioned application range, in this study, we propose a novel, fast, and accurate in-plane displacement distribution measurement method using a digital camera and arbitrary repeated patterns based on the moiré methodology. The key aspect of this method is the use of both the fundamental frequency and the high-order frequency components of the moiré fringe before and after deformations. The principle and primary experimental results are presented, and the effectiveness is demonstrated through a simple displacement measurement. The parameters including the adoption of the highest frequency order, sampling pitch, order of intensity interpolation, and the camera resolution are discussed in regards to their influence on the measurement accuracy.

## 2. Outline of the conventional sampling moiré method for displacement measurements

As is well known, the moiré phenomenon occurs when two repetitive structures are superposed or viewed against each other. Moiré patterns are also produced by digital imaging when a repeated structure is scanned. Recently, we developed an accurate, single-shot phase analysis measurement technique called the sampling moiré method [13]. The basic principle of the sampling moiré method is outlined in Fig. 1.

When a grating with a physical pitch *p* is recorded by a digital camera as shown in Fig. 1(a), the recorded intensity of the grating *f*(*x*, *y*) with an initial phase *φ*_{0} can be given as

*a*and

*b*are the amplitude of the grating intensity and the background intensity, respectively.

*P*is the recorded grating pitch as a pixel unit in the image plane (

*x*,

*y*) of the sensor, and

*φ*is the phase value of the grating that depends on the position of the grating.

By performing down-sampling and linear intensity interpolation [19, 20] as shown in Fig. 1(b), multiple phase-shifted fringe patterns *f _{m}*(

*x*,

*y*;

*k*) can be obtained:

*φ*(

_{m}*x*,

*y*) can be calculated by extracting the fundamental frequency component using a phase-shifting method and a discrete Fourier transform (DFT) algorithm, as presented in Eq. (3).

*φ′*(

_{m}*x*,

*y*) can be determined. Finally, the in-plane displacement amount

*δ*(

*x*,

*y*) is directly obtained from the phase difference of the phase difference distribution Δ

*φ*(

_{m}*x*,

*y*) =

*φ′*(

_{m}*x*,

*y*) −

*φ*(

_{m}*x*,

*y*) of the moiré fringe before and after deformations.

In the sampling moiré method, a regular cosinusoidal or rectangular grating with a constant pitch is usually used. The measurement accuracy is 1/1000 of the grating pitch, and the theoretical error analysis is discussed in detail in Ref. [20].

## 3. Principle of the displacement distribution measurement using repeated patterns

Based on the principle of the sampling moiré method, we propose a novel displacement measurement method that uses a digital camera and arbitrary repeated patterns. The idea of the proposed method is that the fundamental frequency component and the high-order frequency components are used simultaneously to improve the measurement accuracy because any repeated pattern can be represented by a Fourier series with multiple frequencies.

Figure 2(a) illustrates a schematic representation of a periodically repeating pattern, which can be considered the sum of cosines function with multiple frequencies, i.e., a Fourier cosine series. When an image of a repeated pattern is recorded by a digital camera, the intensity of the pattern can be presented as

*a*

^{(w)}is the intensity amplitude of the

*w*-order frequency component in the Fourier spectrum and

*b*is the background intensity.

*W*is the maximum frequency order in the Fourier series.

As in the sampling moiré method, after performing down-sampling and linear intensity interpolation, the multiple phase-shifted moiré fringe patterns *g _{m}*(

*x*,

*y*;

*k*) can be obtained and expressed as

Interestingly, the obtained moiré fringes with low spatial frequencies can also be considered as the sum of cosines functions with multiple frequencies by a Fourier series as shown in Fig. 2(b). The moiré fringes *g _{m}*(

*x*,

*y*;

*k*) contain both the first frequency component and the high-order frequency components. The intensity amplitude

*a*

^{(w)}and the phase distribution of the moiré fringe

*φ*with a frequency

_{m}*w*can be calculated by extracting the

*w*-order frequency component using phase-shifting methods and a DFT algorithm as given in Eqs. (7) and (8).

*w*-order frequency before and after deformations as

To improve the measurement accuracy (signal-to-noise ratio), we use multiple frequency components. Therefore, the final displacement distribution can be obtained using multiple high-order frequency information by considering the amplitude of each frequency component:

where*W*is the highest frequency order to be analyzed for the displacement measurements and

_{h}*â*

^{(w)}(

*x*,

*y*) is the weighting factor for the component with frequency

*w*. In this study, we use five frequency components (

*W*= 5) to determine the final displacement distribution in the experiment.

_{h}Figure 3 summarizes the comparison of the conventional sampling moiré method and the proposed method. In the proposed method, both the first frequency component and the high-order frequency components of the moiré fringe are used to improve the measurement accuracy. Therefore, the measurement accuracy can be improved for any repeated pattern including a nonsinusoidal pattern (sinusoidal pattern with nonlinearity).

## 4. Experiments

To confirm the effectiveness of our proposed method, a simple displacement measurement experiment was performed. The measurement accuracy obtained by the conventional sampling moiré method and the proposed method was compared.

#### 4.1. Experimental setup

The experimental setup is shown in Fig. 4. A digital CCD camera (The Imaging Source, DMK 41BF02.H; Germany) with 1280 vertical pixels and 960 horizontal pixels was used to capture the images. A flat plate with a 10.0 mm pitch rectangle and repeated patterns of alphanumeric characters and a Chinese character (KANJI) was fixed on a linear stage (Suruga Seiki, KX1250C; 1 *μ*m resolution; Japan). These patterns were displaced in the *x* (horizontal) direction from 0 to 1.0 mm in 0.02 mm increments, and a single image was captured at each position. The distance between the CCD camera and the repeated pattern was 1.35 m, and the focus length of the camera lens was 12 mm. In this case, one pitch period of each pattern corresponds to approximately 20 pixels in the recorded image. Therefore, the down-sampling pitch was used as 20 pixels in the displacement analysis, and a linear intensity interpolation was performed.

#### 4.2. Experimental results

Figure 5 shows the experimental results of the in-plane displacement measurement. Figure 5(a) shows the analyzed image with 260×280 pixels. Figures 5(b) and 5(c) show the displacement error distributions obtained by the conventional method and the proposed method, respectively, after a 0.5 mm displacement. Figure 5(d) shows the displacement error obtained by the conventional method in the *y* direction section as a blue line in Fig. 5(b). Figure 5(e) shows the displacement error obtained by the proposed method in the *y* direction section as a red line in Fig. 5(c). In the conventional method, the results introduce a large measurement error in a number of regions, especially for the repeated patterns of “A” and the Chinese characters. However, in our proposed method, the displacement measurement can be performed stably without large measurement errors for all four types of repeated pattern.

Figure 6 shows the experimental results of the root-mean-square (RMS) measurement error in the 20×20 pixels evaluation area (as indicated by the blue area in Fig. 5(a)) for the conventional method and the proposed method for four types of repeated patterns. In the case of the rectangular grating, the average errors were 8.7 *μ*m and 9.6 *μ*m from the conventional method and the proposed method, respectively, as shown in Fig. 6(a). The sampling moiré method measurement is accurate to 1/1000 of the grating pitch, which is similar to our previous study [15]. For the rectangular grating, it seems that the use of only the first frequency is sufficient to measure the displacement with a high accuracy. In the case of the repeated pattern of “3”, the average error was 26.3 *μ*m and 12.1 *μ*m obtained by the conventional method and the proposed method, respectively, as shown in Fig. 6(b). The measurement accuracy was improved 2.2 times. In the case of the repeated pattern of the Chinese characters, the average error was 76.6 *μ*m and 12.2 *μ*m obtained by the conventional method and the proposed method, respectively, as shown in Fig. 6(c). The measurement accuracy was improved 6.3 times. In the case of the repeated pattern of “A”, the average error was 112.4 *μ*m and 10.0 *μ*m obtained by the conventional method and the proposed method, respectively, as shown in Fig. 6(d). The measurement accuracy was improved 11.2 times.

Because the measurement accuracy was on the order of 0.01 mm for all the repeated patterns with a 10 mm pitch, this indicates that the obtainable measurement accuracy is 1/1000 of the pattern pitch; In other words, a 1 nm displacement measurement can be expected when the micro grating pitch is 1 *μ*m. In contrast, a millimeter displacement can be measured if the repeated pattern has a one meter pitch, such as a window of a building, and is captured by a digital camera. These experimental results clearly demonstrated the effectiveness of our proposed method.

## 5. Discussions

Here, the parameters including the adoption of the highest frequency order, sampling pitch, order of intensity interpolation, and the camera resolution are discussed in regards to their influence on the measurement accuracy. In addition, the advantages of our method are listed.

#### 5.1. Influence of the adoption of the highest frequency order on the measurement accuracy

In this study, the key point of the proposed method is the use of both the first frequency and the higher-order frequency components information. Here, the relation between the measurement accuracy and the highest frequency order was investigated.

Figure 7 shows the relation between the highest frequency order (*W _{h}*) and the RMS displacement error for three type repeated patterns after 0.1 mm and 1.0 mm displacements. As shown in Fig. 7(a), the RMS displacement error is decreased when the highest frequency order of the moiré pattern is used. After

*W*becomes greater than 5, the measurement accuracy is nearly identical. As shown in Fig. 7(b), the RMS displacement error is smaller when

_{h}*W*is approximately 4 to 6.

_{h}Figure 8 shows the Fourier spectra for a repeated pattern of the number “3” and the character “A” at the central and lower points, respectively. In the case of the number “3”, the signal mainly appears in the first to third frequency components at the central and lower points. In the case of the character “A”, most of the signal appears in the first to third frequency components at the central point. However, at the lower point, the largest value of the frequency component appears in the second order, and the first frequency component is much smaller. This is the main reason that the measurement accuracy can be dramatically decreased when using the conventional sampling moiré method as shown in Figs. 5(b), 5(d) and Fig. 6(d). In contrast, in our proposed method, the measurement errors of both points are smaller and approximately equal because we consider multiple frequency components as shown in Figs. 5(c), 5(e) and 6(d). From the Fourier spectra in Fig. 8, it appears that the highest frequency order of 5 is sufficient for the displacement analysis.

#### 5.2. Influence of the sampling pitch on the measurement accuracy

To perform accurate displacement measurements, it is important to use an optimum sampling pitch in the sampling moiré method as reported in our early study [15]. We determined that the phase error is introduced by mismatches between the sampling pitch and the grating pitch. The periodic phase error is proportional to the square of the spatial frequency of the moiré fringes for the linear interpolation [20]. Therefore, it is important to use an optimum sampling pitch for the proposed method. To determine the grating pitch as a number of pixels in the recorded image for any repeated periodic pattern, we propose an automatic determination technique as follows.

The local frequency with a Fourier component *w* at the point (*x*, *y*) in the recorded image 1/*p*^{(w)} can be considered to express the phase derivatives as
$\partial {\phi}_{m}^{(w)}(x,y)/\partial x$. Therefore, by differentiating
${\phi}_{m}^{(w)}(x,y)$ with respect to *x* for the phase term of the moiré fringe in Eq. (6), we can obtain the following equation:

*w*. Next, the optimum pitch for down-sampling with frequency

*w*can be obtained as

*P̂*(

*x*,

*y*) can be obtained by considering the amplitude information of multiple frequencies:

In the displacement measurement experiment described in Sec. 4, the grating pitch was determined to be 20.2 pixels in the recorded image. We use 20 pixels as the down-sampling pitch in the displacement analysis. In this case, the displacement error caused by the mismatches between the sampling pitch and the grating pitch can be ignored.

Our method is also useful in analyzing the deformation and strain analysis for material evaluations under tensile/compression testing or thermal deformation measurements. If we can observe a clear periodic pattern, the displacement and strain distributions from small deformations to large deformations can be measured. Within a few percent strain condition, the same sampling pitch for the displacement measurement can be used. When a loaded specimen has a large deformation, the grating pitch can be largely changed. In this case, each optimum sampling pitch of the specimen before and after deformations is preferred to use for the strain measurement.

#### 5.3. Influence of the order of the intensity-interpolation on the measurement accuracy

In the displacement measurement experiment described in Sec. 4, a linear intensity interpolation was used. Based on our recent research on the accuracy of the sampling moiré method [15], if a high-order (2nd- and 3rd-order) interpolation such as when using a *B*-spline function is performed, the phase error caused by the mismatch between the sampling pitch and the grating pitch can be decreased compared to a linear interpolation. In contrast, in the case of higher-order interpolations, the spatial resolution is lower than that for a linear interpolation. The relation between the spatial resolution *N* and the order of the intensity interpolation *O* is given as

To investigate the effect of the mismatch between the sampling pitch and the grating pitch, the displacement measurement was also analyzed using a sampling pitch *T* = 18 pixels. Figure 9 shows the Fourier spectra of the repeated pattern for character “A” at the central point when the order of the intensity interpolation is changed from 1st-order (linear) to 3rd-order for sampling pitches of *T* = 20 pixels and *T* = 18 pixels, respectively. In the case of *T* = 20 pixels (i.e., a sampling pitch coincident with the grating pitch), the Fourier spectra for three different interpolation methods are nearly identical as shown in Fig. 9(a). In the case of *T* = 18 pixels (i.e., the sampling pitch is approximately equal to grating pitch), the Fourier spectra for three different interpolation methods are slightly different as shown in Fig. 9(b).

Figure 10 shows the relation between the order of the intensity interpolation and the RMS displacement error obtained by the sampling moiré method and the proposed method in the case of *T* = 20 pixels and *T* = 18 pixels, respectively, after a 0.1 mm displacement. The RMS displacement error is obtained in the central 20×20 pixels evaluation area (400 points) for the repeated pattern for the character “A”. For the conventional sampling moiré method, as shown in Fig. 10(a), the measurement accuracy obtained by the higher-order interpolation is better than the linear interpolation because a larger number of pixels of the original grating are required to analyze the phase value of the moiré fringe at a single pixel. When the sampling pitch does not match the grating pitch, the RMS measurement error can be increased, especially for the linear interpolation as shown in Fig. 10(b).

In contrast, the RMS displacement error obtained by the proposed method was nearly identical for the three different interpolation methods as shown in Figs. 10(a) and 10(b). This indicated that the order of the intensity interpolation is not an important factor in the measurement accuracy in the proposed method. Because the algorithm and implementation of the linear interpolation is much simpler, the calculation speed is higher than the high-order interpolation. From the results of Fig. 10, it can be considered that the linear interpolation is sufficient in the displacement analysis for the proposed method.

#### 5.4. Influence of the camera resolution on the measurement accuracy

In our proposed method, after performing down-sampling and an intensity interpolation, the phase distribution of the moiré fringe is determined by the well-known phase-shifting method. In the *N*-step phase-shifting algorithm, the relation between the standard deviation of the phase error *σ _{φn}* and the standard deviation of the random noise

*σ*is as follows:

_{n}Clearly, to improve the measurement accuracy, we can increase the signal-to-noise ratio, i.e., by recording a high-contrast fringe pattern at low-noise conditions. The other method is to increase the number of phase-shifting steps *N*, which corresponds to using a high-resolution camera and performing an analysis using a large value of the sampling pitch. As indicated in Eq. (16), if an image with twice the resolution is used, the phase error caused by random noise can be reduced by a factor of
$\sqrt{2}$.

#### 5.5. Advantages

Compared with existing displacement sensors and measurement techniques, the main advantages of the proposed method are as follows:

*Full-field*: Compared with point-by-point sensors or methods, the proposed method can measure the displacement distribution of the measured object.*Accuracy*: The displacement distribution is analyzed by using multiple frequency components of the moiré fringe, and the measurement accuracy achieved is 1/1000 of the pattern pitch regardless of the type of repeated patterns.*Fast*: The proposed method only requires each image before and after deformations. Therefore, this method can be easily applied to dynamic conditions by using a high-speed camera.*Easy setup*: This method does not require a complicated optical setup. We simply record the digital images during deformations.*Wide applicability*: This method can be useful for various applications, including in materials science at the nano-scale, in the biomimetics field at the microscopic scale, and for health monitoring of infrastructures at macroscopic scale if any repeated patterns exist on thier surface.

## 6. Conclusions

This paper presented a fast and accurate in-plane displacement distribution measurement technique that utilizes an arbitrary repeated pattern and the moiré methodology. Based on the basic principle of the conventional sampling moiré method, the proposed method utilizes the first frequency component as well as multiple high-order frequency components to analyze the displacement distribution of the moiré fringe generated by the repeated pattern. Because multiple phase information including high-order frequency components are used, the displacement measurement can be performed accurately regardless of the type of repeated pattern.

The effectiveness was confirmed through an in-plane displacement measurement experiment using several repeated patterns, including alphanumeric characters and Chinese characters as examples. The experimental results demonstrated that small displacement can be measured to an accuracy of 1/1000 of the pattern pitch. This method is useful to various applications ranging from the study of displacement and strain distributions in materials science, the biomimetics field, and in mechanical material testing, to secure the integrity of infrastructures.

## Acknowledgments

This work is supported by JSPS KAKENHI, Grant-in-aid for Young Scientists (B), Grant No. 24760095.

## References and links

**1. **M. J. Hÿtch, J-L. Putaux, and J-M. Pénisson, “Measurement of the displacement field of dislocations to 0.03Å by electron microscopy,” Nature **423**, 270–273 (2003). [CrossRef]

**2. **J. J. Lee and M. Shinozuka, “A vision-based system for remote sensing of bridge displacement,” NDT&E Int **39**, 425–432 (2006). [CrossRef]

**3. **J. M. Burch and C. Forno, “A high sensitivity moiré grid technique for studying deformation in large objects,” Opt. Eng. **14**, 178–185 (1975). [CrossRef]

**4. **S. Avril, A. Vautrin, and Y. Surrel, “Grid method: application to the characterization of cracks,” Exp. Mech. **44**, 37–43 (2004). [CrossRef]

**5. **D. Post, B. Han, and P. Ifju, *High Sensitivity Moiré: Experimental Analysis for Mechanics and Materials* (Springer-Verlag, 1994), Chap.4. [CrossRef]

**6. **B. Han and Y. Guo, “Thermal deformation analysis of various electronic packaging products by moiré and microscopic moiré interferometry,” J. Electron. Packaging, Trans. ASME **117**, 185–191 (1995). [CrossRef]

**7. **S. Kishimoto, M. Egashira, and N. Shinya, “Observation of micro-deformation by moiré method using a scanning electron microscope,” J. Soc. Mat. Sci. **40**, 637–641 (1991). [CrossRef]

**8. **H. Xie, Q. Wang, S. Kishimoto, and F. Dai, “Characterization of planar periodic structure using inverse laser scanning confocal microscopy moiré method and its application in the structure of butterfly wing,” J. Appl. Phys. **101**,103511 (2007). [CrossRef]

**9. **S. Kishimoto, M. Egashira, and N. Shinya, “Microcreep deformation measurements by a moiré method using electron beam lithography and electron beam scan,” Opt. Eng. **32**, 522–526 (1993). [CrossRef]

**10. **M. Tang, H. Xie, J. Zhu, X. Li, and Y. Li, “Study of moiré grating fabrication on metal samples using nanoimprint lithography,” Opt. Express **20**, 2942–2955 (2012). [CrossRef] [PubMed]

**11. **Q. Wang, S. Kishimoto, Y. Tanaka, and Y. Kagawa, “Micro/submicro grating fabrication on metals for deformation measurement based on ultraviolet nanoimprint lithography,” Optics and Lasers in Engineering **51**, 944–948 (2013). [CrossRef]

**12. **S. Kishimoto, Y. Tanaka, T. Tomimatsu, Y. Kagawa, and K. Nagai, “Fabrication of micromodel grid for various moiré methods by femtosecond laser exposure,” Opt. Lett. **34**, 112–114 (2009). [CrossRef] [PubMed]

**13. **S. Ri, M. Fujigaki, and Y. Morimoto, “Sampling moiré method for accurate small deformation distribution measurement,” Exp. Mech. **50**, 501–508 (2010). [CrossRef]

**14. **M. Fujigaki, K. Shimo, A. Masaya, and Y. Morimoto, “Dynamic shape and strain measurements of rotating tire using a sampling moiré method,” Opt. Eng. **50**,101506 (2011). [CrossRef]

**15. **S. Ri, T. Muramatsu, M. Saka, K. Nanbara, and D. Kobayashi, “Accuracy of the sampling moiré method and its application to deflection measurements of large-scale structures,” Exp. Mech. **52**, 331–340 (2012). [CrossRef]

**16. **S. Ri, M. Saka, K. Nanbara, and D. Kobayashi, “Dynamic thermal deformation measurement of large-scale, high-temperature piping in thermal power plants utilizing the sampling moiré method and grating magnets,” Exp. Mech. **53**, 1635–1646 (2013). [CrossRef]

**17. **Q. Wang, S. Kishimoto, X. Jiang, and Y. Yamauchi, “Spot moiré fringes: determination of domain boundaries and structural parameters in ordered nanoporous structures,” Chem. Eur. J. **20**, 2179–2183 (2014). [CrossRef]

**18. **K. Patorski, M. Wielgus, M. Ekielski, and P. Kaźmierczak, “AFM nanomoiré technique with phase multiplication,” Meas. Sci. Technol. **24**,035402 (2013). [CrossRef]

**19. **Y. Arai, S. Yokozeki, K. Shiraki, and T. Yamada, “High precision two-dimensional spatial fringe analysis method,” J. Mod. Opt. **44**, 739–751 (1997). [CrossRef]

**20. **S. Ri and T. Muramatsu, “Theoretical error analysis of the sampling moiré method and phase compensation methodology for single-shot phase analysis,” Appl. Opt. **51**, 3214–3223 (2012). [CrossRef] [PubMed]