General spatial phase-shifting interferometry by optimizing the signal retrieving function

Yi Wang; Xiang Qiu; Jiaxiang Xiong; Bingbo Li; Liyun Zhong; Shengde Liu; Yunfei Zhou; Jindong Tian; Xiaoxu Lu

doi:10.1364/OE.25.007170

1. Introduction

In optical off-axis interferometry system, the phase distribution induced by the tested sample can be retrieved from only one-frame interferogram [1] and the corresponding system configuration is simpler and more robust compared with the PZT-driven temporal phase-shifting methods or other multiplexing methods such as the polarization encoded technology [2, 3]. Thus, the off-axis interferometry should be a better candidate in dynamic phase measurement [4–8]. Generally speaking, the phase retrieval solutions from one-frame interferogram with spatial carrier can be classified as: the global transformation method [9–21] and the spatial phase-shifting (SPS) method [8, 22–28]. All of these methods are based on sacrificing the spatial bandwidth of interferogram to gain more information about the phase signal.

In Fourier transform (FT) method, the phase retrieval is achieved by peaks separation in frequency domain. After the + 1 level signal peak is retrieved, we can achieve the modulation phase through the inverse FT. One main disadvantage of FT method is the Gibbs effect [12, 14, 15] which is induced by the wide-band modulation of spatial-frequency components. These errors is apparent near the locations with phase jump and will be spread out in the spatial domain due to the global property of FT, and this will lead to the lack of local calculation property. To address this, the SPS method is introduced through using the local calculation property.

In SPS method, a sequence of sub-interferograms are constructed from one-frame carrier interferogram by moving an image-captured window along the transverse and longitudinal directions, respectively. In general, there are phase shifts among these sub-interferograms due to the large spatial carrier, so the phase-shifting algorithm can be utilized to perform phase retrieval.

Typically, there is a well-known compromised solution in SPS method: the more sub-interferograms are constructed, the stronger noise-resistant ability is achieved and the more spatial bandwidth is sacrificed [28]. And this is a direct consequence of its pixels-division multiplexing property. Along with the number increasing of sub-interferograms, the effective density of pixel array is decreased. Therefore, it is important to construct and utilize appropriate number of sub-interferograms. However, in conventional SPS methods, one important character of off-axis interferogram with spatial carrier, revealing the intensity variation of background term is much slower than signal modulation term, is not effectively utilized. Based on the local property of SPS method, the background term is allowed to have the same variation rate with the signal modulation term due to the calculation is performed from the same pixels of one group constructed sub-interferograms. This will make the intensity oscillation induced by the signal-modulation term be partially coupled into the background term. To suppress this kind of error and reserve local calculation property simultaneously to improve the accuracy of phase retrieval, we propose a new method named as general spatial phase-shifting (GSPS) in this study. Following, we will introduce the proposed method in details.

2. Principle

One-frame interferogram with spatial carrier can be mathematically described as

I (x, y) = A (x, y) + B (x, y) \cos [ϕ (x, y) + f_{0 x} x + f_{0 y} y] + n (x, y) .

in which x and y denote the Cartesian coordinates in CCD target,

A (x, y)

and

B (x, y)

represent the background and modulation amplitude in interferogram, respectively;

f_{0 x}

and

f_{0 y}

represent the spatial carrier along the x and y directions,

ϕ (x, y) + f_{0 x} x + f_{0 y} y

and

n (x, y)

are the phase distribution and noise distribution of interferogram, respectively.

Next, N-frame phase-shifting sub-interferograms are constructed from above one-frame carrier interferogram by moving an image-captured window along the transverse and longitudinal directions, respectively

{\begin{cases} I_{i} = A (x, y) + B (x, y) \cos [ϕ (x, y) + f_{0 x} x + f_{0 y} y + δ_{i}] + n_{i} (x, y) + d_{i} (x, y) \\ = A (x, y) + B (x, y) \cos [ϕ' (x, y) + δ_{i}] + o_{i} (x, y) \\ i = 1... N \end{cases}

in which

ϕ' (x, y) = ϕ (x, y) + f_{0 x} x + f_{0 y} y

;

n_{i} (x, y)

represents the intensity fluctuation of noise,

d_{i} (x, y)

represents the deviation between the approximation mathematical model and the practical condition, which is mainly induced by the rapid phase variation of local areas;

δ_{i}

denotes the phase-shifts . Here, we define total error as

o_{i} (x, y) = n_{i} (x, y) + d_{i} (x, y)

.

In SPS method, the corresponding $B (x, y) \cos ϕ (x, y)$ and $B (x, y) \sin ϕ (x, y)$ can be expressed as

{\begin{cases} B (x, y) \cos ϕ' (x, y) = \sum_{i} c_{i} I_{i} = c (x, y) \otimes I (x, y) \\ B (x, y) \sin ϕ' (x, y) = \sum_{i} c_{i}' I_{i} = c' (x, y) \otimes I (x, y) \\ i = 1...... N \end{cases} .

Without the loss of generality, it is assumed that four-frame sub-interferograms are achieved with the spatial-carrier induced phase shifts of

0, \frac{π}{2}, π, \frac{3 π}{2}

. By employing four-step phase-shifting algorithm, we have that

{\begin{cases} B (x, y) \cos ϕ' (x, y) = I_{1} - I_{3} = c (x, y) \otimes I (x, y) \\ B (x, y) \sin ϕ' (x, y) = I_{4} - I_{2} = c' (x, y) \otimes I (x, y) \end{cases} .

in which

c, c'

can be respectively expressed as

{\begin{cases} c = [δ (x, y) - δ (x - 2 x_{0}, y)] / 2 \\ c' = [δ (x - 3 x_{0}, y) - δ (x - x_{0}, y)] / 2 \end{cases} .

Here, we have assumed that that the direction of spatial carrier is along only x axis, and

x_{0}

represents the pixel interval, and then we can construct a new function

h (x, y) = c (x, y) + i c' (x, y) .

Thus, there is a relationship

B (x, y) \exp [- i ϕ' (x, y)] = h (x, y) \otimes I (x, y) .

in which

h (x, y)

denotes the filtering function employed for phase retrieval in four-step SPS method, and the corresponding distribution in spatial frequency domain can be expressed as

H (f_{x})

, and the corresponding distribution in frequency domain can be expressed as

H (f_{x})

(Here, for simplicity, the variation of

h (x, y)

along y has been neglected)

{\begin{cases} ξ [B (x) \exp - i ϕ' (x)] = H (f_{x}) ξ [I (x)] \\ H (f_{x}) = ξ [h (x)] = \frac{1 - \exp (i 4 π x_{0} f_{x}) - i \exp (i 2 π x_{0} f_{x}) + i \exp (i 6 π x_{0} f_{x})}{2} \end{cases} .

Here,

ξ

denotes Fourier transform operator and

x_{0} = 1 / T_{f}^{}

, in which

x_{0}

and

T_{f}

are the spatial interval of the image detector and the period of spatial spectrum, respectively. Then the spatial carrier can be expressed as

f_{x 0} = T_{f} / 4

. Following, we calculate the value of

H (f_{x})

at three important frequency points of

f_{x} = - T_{f} / 4, 0, T_{f} / 4

, we have that

{\begin{cases} H (- T_{f} / 4) = H (0) = 0 \\ H (T_{f} / 4) = 2 \end{cases} .

Equation (9) indicates that

H (f_{x})

can retrieve the + 1 signal peak locating at

T_{f} / 4

and eliminate the −1 peak at

- T_{f} / 4

and the 0 peak at central in the case that these spectral peaks are not extended. Otherwise, the wrong coupling error (WCE) induced by the non-zero modulation between

H (f_{x})

and the 0 peak, the −1 peaks will appear. Figure 1 shows the curve of both the imaginary part and real part of

H (f_{x})

. We can see that the background term has higher intensity relative to the signal modulation term, so the WCE induced by the 0 peak is larger than that the −1 peak. Especially, if the spatial carrier value becomes smaller, the WCE induced by the 0 peak will become larger due to the oscillation of

H (f_{x})

near the zero frequency is increased with the spatial carrier reduction. This is easy to understand because if the spatial carrier becomes small, the validity of the mathematical approximation assumed in SPS method will be weak.

Fig. 1 Frequency spectrum distribution of one-frame carrier interferogram, in which the filtering function cannot totally eliminate the 0 and −1 peaks, so the retrieved signal will contain the WCE induced by the 0 peak and −1 peak).

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Although the signal retrieving function $H (f_{x})$ of SPS method is not perfect, the advantage of SPS method is still apparent: Due to the irrelevant information from far-zone can be effectively excluded, the result achieved from the SPS method with local calculation is more reliable than FT method with global transformation. What’s more, the local error will no longer be spatially spread out in FT method. Noted that suppressing the WCE and improve the performance of SPS method, we need to modify $H (f_{x})$ . The WCE induced by the 0 peak is more significant than the −1 peak, so we can regardless the WCE induced by the −1 peak which has a strong relationship with the local property of $H (f_{x})$ and simply modify it by improving its low frequency rejection property to suppress the WCE from 0 peak. Usually, this procedure can be simplified by pre-filtered the 0 peak with a high-pass filter before the local calculation of SPS method. This is a narrow-band, low-frequency modification of $H (f_{x})$ without a wide-band spectrum modulation adopted in FT method, thus it almost has no negative affect to the local calculation property. It can be mathematically described as

\begin{array}{l} I (f_{x}, f_{y}) = ξ [I (x, y)] = A (f_{x}, f_{y}) + ξ [B (x, y) \cos ϕ' (x, y)] \\ \Rightarrow I' (f_{x}, f_{y}) = G (f_{x}, f_{y}) I (f_{x}, f_{y}) = ξ [B (x, y) \cos ϕ' (x, y)] . \\ \Rightarrow I' (x, y) = ξ^{- 1} [I' (f_{x}, f_{y})] = B (x, y) \cos ϕ' (x, y) \end{array}

Therefore, by employing a low-frequency suppressed filtering function

G (f_{x}, f_{y})

, we can keep the local property of SPS method and suppressing the WCE from background term. This is a key step of GSPS method.

As we know, in SPS method, it is needed to determine the value of spatial carrier or the phase shifts between the constructed sub-interferograms. Usually, it is difficult to fix the value of spatial carrier in experiment, so the self-calibrating algorithm is introduced [27, 28], in which the advanced iterative algorithm(SPS-AIA) reveals high accuracy while its speed is low due to the iterative procedure [27]. In this study, the iterative procedure is similar with the SPS-AIA method, so the proposed method is named as the GSPS-AIA method. The main steps of the proposed GSPS-AIA algorithm are as following:

After the background term A(x,y) is filtered, the sub-interferograms can be expressed as

{\begin{cases} I_{i}' = B (x, y) \cos [ϕ (x, y) + f_{0 x} x + f_{0 y} y + δ_{i}] + o_{i} (x, y) \\ = B (x, y) \cos [ϕ' (x, y) + δ_{i}] + o_{i} (x, y) \\ i = 1...... N \end{cases} .

First, we estimate a group of phase shifts

δ_{1} ...... δ_{N}

to start the iteration, and then calculate the phase by using the least-square fitting algorithm

\begin{array}{l} L_{f} = (\begin{matrix} \sum_{i = 1}^{i = N} \cos^{2} δ_{i} & \sum_{i = 1}^{i = N} \cos δ_{i} \sin δ_{i} \\ \sum_{i = 1}^{i = N} \cos δ_{i} \sin δ_{i} & \sum_{i = 1}^{i = N} \sin^{2} δ_{i} \end{matrix}) \\ Λ_{f} = {\sum_{i = 1}^{i = N} I_{i}' (x, y) \cos δ_{i}, \sum_{i = 1}^{i = N} I_{i}' (x, y) \sin δ_{i}} \\ X_{f} = {B (x, y) \cos ϕ' (x, y), - B (x, y) \sin ϕ' (x, y)} . \end{array}

And there is a relationship in Eq. (12)

{\begin{cases} X_{f} = L_{f}^{- 1} Λ_{f} \\ ϕ' (x, y) = \arctan (- X_{f} (2) / X_{f} (1)) \end{cases} .

where i denotes the index of sub-interferograms.

Second, we substitute the $ϕ' (x, y)$ from first step back into Eq. (11) and assume that $B (x, y)$ is a spatial irrelevant constant $b$ , then the phase shifts between sub-interferograms can be also achieved through using the least square algorithm

\begin{array}{l} L_{f}' = (\begin{matrix} \sum_{j = 1}^{j = M} \cos^{2} ϕ'_{j} & \sum_{j = 1}^{j = M} \sin ϕ'_{j} \cos ϕ'_{j} \\ \sum_{j = 1}^{j = M} \sin ϕ'_{j} \cos ϕ'_{j} & \sum_{j = 1}^{j = M} \sin^{2} ϕ'_{j} \end{matrix}) . \\ Λ_{f}' = {\sum_{j = 1}^{j = M} I_{j}' \cos ϕ'_{j}, \sum_{j = 1}^{j = M} I_{j}' \sin ϕ'_{j}} \\ X_{f}' = {b \cos δ_{i}, b \sin δ_{i}} \end{array}

in which

{\begin{cases} X_{f}' = L_{f}'^{- 1} Λ_{f}' \\ δ_{i} = \arctan (- X_{f}' (2) / X_{f}' (1)) \end{cases} .

In Eqs. (14), j denotes the index of the pixels array, in which the 2-dimmensional pixels array is rearranged as the 1-dimmensional pixels array.

Third, by substituting the phase shifts $δ_{i}$ back to Eq. (11), one turn of iterative calculation is completed. When the condition that $\max | δ_{i + 1} - δ_{i} | < ε$ is satisfied, the iterative procedure is ended and the final phase distribution $ϕ'$ can be determined, here, $ε$ denotes a small threshold value. Note that the size of $L_{f}$ and $L_{f}'$ is reduced from $3 \times 3$ pixels in SPS-AIA method [27, 29] to $2 \times 2$ pixels in the proposed GSPS-AIA method, this will greatly improved the calculation speed.

Although in the second step of iterative procedure, both the modulation amplitude $B (x, y)$ and background $A (x, y)$ are assumed to be constant, but in the practical application, if the fringe number is large enough, the accuracy of phase retrieval is insensitive to the nonuniformity of these two parameters. On the other hand, for each pixel, the difference of $B (x, y)$ between different sub-interferograms will lead to the error in both the proposed method and conventional SPS method, but compared with the error induced by the spatial-dependence of encoded phase, the error induced by $B (x, y)$ and $A (x, y)$ is very small and can be neglected.

As we know, the initial phase shifts $δ_{i}$ will significantly affect the iterative number, and to decrease the calculation time, it is needed to perform the estimation of $δ_{i}$ before the iterative calculation. Based the idea of reference [30], in this study, we perform the prior estimation of spatial phase shifts as following: By introducing 3-frame sub-interferograms $I'_{1}, I'_{2}, I'_{3}$ , in which $I_{2}'$ and $I_{3}'$ are respectively constructed by moving one pixel of $I_{1}'$ along the right and down directions, and the corresponding spatial phase shifts $δ_{x}$ and $δ_{y}$ can be determined by

{\begin{cases} δ_{x} = \arccos (\frac{\sum_{x, y} I_{1}' \cdot I_{2}'}{\sum_{x, y} I_{1}' \cdot I_{1}'}) \\ δ_{y} = \arccos (\frac{\sum_{x, y} I_{1}' \cdot I_{3}'}{\sum_{x, y} I_{1}' \cdot I_{1}'}) \end{cases} .

Thus, the phase shifts can be estimated from Eq. (16), and then we can utilize these to start the iterative calculation described by Eq. (12) to Eq. (15) .

Till now, we have clarified the basic principle of the proposed GSPS-AIA method. Next, we will verify its performance by the simulation and experimental research.

3. Numerical simulation

One-frame off-axis carrier fringe pattern is generated, in which the size of interference pattern is set as $400 \times 400$ pixels with the interval of 0.01mm; the background and modulation are $A (x, y) = 50 \exp [- 0.15 (x^{2} + y^{2})] + 50$ and $B (x, y) = 75 \exp [- 0.15 (x^{2} + y^{2})]$ , respectively; the phase distribution and the spatial carrier along x and y directions are $ϕ (x, y) = 2 p e a k s (500) + f_{x 0} x + f_{y 0} y, f_{x 0} = 80, f_{y 0} = 80$ . In addition, the noise with SNR of 30db is added to the fringe pattern. The phase shifts of sub-interferograms in SPS-AIA method are set as 0rad,0.1rad,0.15rad,0.2rad,0.25rad,0.3rad,0.35rad,0.4rad,0.45rad. Considering that the SPS-PCA method reveals fast speed and high accuracy, we also perform the comparison between the proposed method and SPS-PCA method. In addition, it is reported that the SPS method has better accuracy if the number of sub-interferograms N is equal to 9 [28, 31]. Then we choose 9-frame sub-interferograms to perform phase in both the GSPS-AIA method and other comparison methods in this study. Figures 2(a) and 2(b) give one-frame simulated interference pattern and the corresponding theoretical phase distribution, respectively; Figs. 2(c) to 2(e) show the retrieved phases by the GSPS-AIA, SPS-AIA and SPS-PCA methods, respectively; Fig. 2(f) to 2(h) show the differences between the theoretical phase and the phase achieved with three different methods; Figs. 2(i) to 2(k) present the plane view of Fig. 2 (f) to Fig. 2 (h), respectively, in which a personal computer with CPU of Intel i7 2.8 GHZ is utilized for calculation.

Fig. 2 (a) One-frame simulated carrier interference pattern; (b) the theoretical phase distribution of (a); the plane view of the achieved phase by using (c) GSPS-AIA method with RMSE of 0.029 rad, (d) SPS-AIA method with RMSE of 0.048 rad; (e) SPS-PCA method with RMSE of 0.048 rad; the difference between the theoretical phase and the phase achieved with (f) GSPS-AIA method; (g) SPS-AIA method; (h) SPS-PCA method; The plane view of error distribution with (i) GSPS-AIA method; (j) SPS-AIA method;(k) SPS-PCA method.

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We can see that the difference between the theoretical phase and the phase achieved with the SPS-AIA and SPS-PCA method is apparently larger than the GSPS-AIA method, indicating the GSPS-AIA method can more effectively suppress the WCE compared with the other two methods. From Fig. 2(j) and Fig. 2(k), it is also found that using the SPS-AIA and SPS-PCA methods, the large error is observed around the location with the large phase gradient, but in Fig. 2(i), this kind of error is effectively suppressed, indicating that the GSPS-AIA method with low WCE can work better than the other SPS methods when the ratio of the spatial carrier to phase gradient is small. Table 1 shows the RMSE, PVE and calculation time respectively achieved with the GSPS-AIA, SPS-AIA and SPS-PCA methods. It is observed that both the speed and accuracy of phase retrieval with the proposed GSPS-AIA method have obvious advantages.

Table 1. RMSE, PVE and calculation time achieved with different methods

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Subsequently, Fig. 3 presents the variation curves of RMSE of the retrieved phase in different spatial carrier value and different noise level. In Fig. 3 (a), it is assumed that the SNR is fixed to 30db, and the direction of spatial carrier is along the diagonal in which $f_{0 x} = f_{0 y}$ with the variation from 0.55 to 2.45rad/pixel. The larger spatial carrier is not suitable for the practical measurement due to the pixel size cannot be neglected when the interference fringe is of high density, then they can no longer be regarded as an array of discrete points. Actually, the fringe contrast will be decreased as the increasing of spatial carrier. If we denote the size of pixels as $x_{0}$ , and the spatial integrated intensity detected by one pixel can be described as (for simplicity, the intensity variation along y has been neglected)

\begin{array}{l} I (x) = A (x) + \frac{\int_{- x_{0} / 2}^{x_{0} / 2} B (x') \cos (ϕ_{0} + f_{x} x' + f_{x} x) d x'}{x_{0}} . \\ = A (x) + \frac{2 \sin (f_{x} x_{0} / 2)}{f_{x} x_{0}} B (x) \cos (ϕ_{0} + f_{x} x) \end{array}

in which the modulation factor

\frac{2 \sin (f_{x} x_{0} / 2)}{f_{x} x_{0}}

will lead to the reduction of fringe contrast . Therefore, we set the simulated spatial carrier along x and y directions as the above mentioned range. In Fig. 3(b), the spatial carrier is fixed at

f_{0 x} = f_{0 y} = 0.6

rad/pixel, and the noise level is changed from 26db to 60db.

Fig. 3 (a) Variation of RMSE of the retrieved phase (a) in different spatial carrier; (b) in different SNR.

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From Fig. 3(a), we can see that if the spatial carrier is large, the accuracy of retrieved phase with all three methods is high due to the ratio of the spatial carrier to the phase variation rate is large. In this situation, $H (f_{x})$ has relative small value near the 0 peak and the corresponding WCE induced by the background term is low. However, if the spatial carrier become smaller, we can see that the RMSEs with the SPS-AIA and SPS-PCA methods will greatly increase but maintain unchanged with the GSPS-AIA method. This is because when the spatial carrier becomes smaller, the WCE induced by the background term will become larger, which is coincided with the above theoretical analysis. Importantly, this result further demonstrates that the proposed GSPS method is very suitable for the phase measurement with rapid variation. In Fig. 3(b), it is also presented that the GSPS method has stronger anti-noise ability relative to the SPS method.

In the above discussion, though we only present the result achieved by combining the GSPS method with AIA method. Actually, by combining the GSPS method with all other SPS methods, we can improve the corresponding accuracy of phase retrieval through improving the low frequency rejection property of signal retrieving function. Next, we will prove the validity of the proposed method by the experiment.

4. Experimental result

Next, the performance of the proposed GSPS-AIA method is verified by the experimental research. A classical transmission mode Mach-Zhnder off-axis microscopic interferometer is built, as illustrated in Fig. 4. Two objectives with magnification of 25 $\times$ and NA of 0.5 are employed; A He-Ne frequency stabilized laser with wavelength of 632.8nm is utilized as the light source, and a guide plate of mobile phone is chosen as the measured sample, in which many subovate transparent dots is embedded into the surface; the resolution of this microscopic system is estimated by $D = \frac{0.61 λ}{N A} \approx 770 n m$ ; the scale bar is determined by a standard resolution testing board placed on the sample. For comparison, 140-frame temporal phase-shifting interferograms with size of 280 $\times$ 270 pixels are employed to calculate the reference phase. Then, we choose one-frame of these phase-shifting interferograms to perform the phase retrieval through using the GSPS-AIA, SPS-AIA and SPS-PCA methods, as shown in Fig. 5. The spatial carrier is 0.577rad/pixel in x direction and −1.048rad/pixels in y direction.

Fig. 4 Experimental setup of the Mach-Zhnder type phase-shifting system.

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Fig. 5 (a) One frame experimental carrier interferogram of a subovate transparent dot in guide plate with the resolution about 770nm; (b) the reference phase achieved from 140-frame temporal phase-shifting interferograms with AIA method; the differences between the reference phase and the phase achieved with different methods (c) GSPS-AIA; (d) SPS-AIA; (e) SPS-PCA; (f) the difference between the reference phase and the retrieved phases of the 60th row

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Table 2 shows the PVE, RMSE and calculation time of phase retrieval with the SPS-AIA, GSPS-AIA and SPS-PCA methods, respectively. We can see that both the accuracy and speed of phase retrieval with GSPS-AIA method reveals obvious advantages relative to the SPS-PCA and the SPS-AIA methods. This experimental result further demonstrates the outstanding performance of the proposed GSPS method.

Table 2. RMSE, PVE and calculation time achieved with different methods

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5. Summary

In summary, we propose a new GSPS method to achieve phase retrieval from one-frame carrier frequency interferogram. By optimizing the signal retrieving function of SPS method, the WCE induced by the background term can be effectively suppressed, and then the accuracy, anti-noise ability and speed of phase retrieval can be greatly improved. Meanwhile, by reserving the local calculation property of SPS method, in the case that the spatial carrier is small, the proposed GSPS method reveals the outstanding performance in both the accuracy. Thus, using the proposed method, even without high-density interference fringe, the more details of measured sample can be reserved. In addition, by combining with the iterative self-calibrating SPS algorithm, such as AIA, the calculation time can be further improved, and this will greatly facilitate more application of proposed method.

Funding

National Natural Science Foundation of China (NSFC) (61475048, 61275015 and 61177005).

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Methods	SPS-AIA	GSPS-AIA	SPS-PCA
PVE(rad)	0.2430	0.1462	0.2505
RMSE(rad)	0.0475	0.0289	0.0475
Time(s)	2.980	0.132	0.100

Method	SPS-AIA	GSPS-AIA	SPS-PCA
PVE(rad)	0.867	0.789	0.862
RMSE(rad)	0.152	0.109	0.152
Time(s)	1.222	0.053	0.033

Methods	SPS-AIA	GSPS-AIA	SPS-PCA
PVE(rad)	0.2430	0.1462	0.2505
RMSE(rad)	0.0475	0.0289	0.0475
Time(s)	2.980	0.132	0.100

Method	SPS-AIA	GSPS-AIA	SPS-PCA
PVE(rad)	0.867	0.789	0.862
RMSE(rad)	0.152	0.109	0.152
Time(s)	1.222	0.053	0.033

General spatial phase-shifting interferometry by optimizing the signal retrieving function

Abstract

1. Introduction

2. Principle

3. Numerical simulation

4. Experimental result

5. Summary

Funding

References and links

Cited By

Figures (5)

Tables (2)

Equations (17)

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