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Matrix p- norm for phase retrieval under generalized three-step phase-shifting and its application on a red blood cell
Optik - International Journal for Light and Electron Optics 178 (2019) 1154–1162
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Optik journal homepage: www.elsevier.com/locate/ijleo
Original research article
Matrix p- norm for phase retrieval under generalized three-step phase-shifting and its application on a red blood cell
T
⁎
Yawei Wanga,b, , Hao Hanb, Yuanyuan Xua, Ying Jia a b
Faculty of Science, Engineering, Jiangsu University, Zhenjiang, 212013, China School of Mechanical Engineering, Jiangsu University, Zhenjiang, 212013, China
A R T IC LE I N F O
ABS TRA CT
Keywords: Generalized phase-shifting interferometry Matrix p- norm Red blood cell Phase retrieval
Phase-shifting interferometry (PSI) is an important technique, which has been widely used in optical measurement and microscopy. Based on the matrix p- norm algorithm, a phase retrieval method applied to the generalized three-step PSI is proposed in this paper. In this method, only three phase-shifted interferograms are required without anyother measurements. When the norm p is an even number, we can obtain the good results. As for the experiment, the setup is simple in structure and easy for experimental operation. The result of a plane wavefront suggestes its accuracy. Both simulation and experimental results of a red blood cell verify that the effectiveness and practicability of this method.
1. Introduction As an important technical mean for optical measurement and microscopy, phase-shifting interferometry (PSI) has widely used [1–14]. In traditional PSI, a special and constant phase shift in each frame or the equal phase shift between two adjacent frames is often required. However, the actual value of the phase shift is slightly different from its theoretical value due to some errors in practice. In order to settle such a serious problem, many methods have been reported [15–23], among which, the generalized phase shift exaction techniques provide a possibility for extracting arbitrary unknown phase shifts from holograms directly. For example, the research group of Fei [15] applied the iterative algorithm to extract the phase shift of each single-wavelength from simultaneous phase-shifting multi-wavelength interferograms. Wang [16] proposed an advanced iterative algorithm based on the least squares method to determine the phase shift and the phase distribution simultaneously. Comparing with the methods mentioned above, the following ones have attracted more attention for the less computational loads and rapid processing rate. Cai and other researchers proposed a class of statistical algorithms to calculate the unknown phase shift directly and reconstruct the wavefront from two or more holograms [17,18]. In these algorithms, the statistical property of the diffraction field of the measured object is required to be considered. Besides, some other measurements, such as the intensity of reference wave, are also needed. Arming at the limitation of the statistical property, Xu [19] proposed an advanced method to extract the phase shift accurately, according to the histogram of phase difference between two adjacent frames. However, it is required to determine both the background intensity and the modulation amplitude by searching for the maximum and the minimum intensity in each pixel of interferograms. As a result, this method is very time-consuming. And then an asynchronous phase-shifting method based on the principal component analysis (PCA) is proposed with high precision and rapid speed, but it is required to eliminate the background term by a temporal average in advance
⁎
Corresponding author at: Faculty of Science, Jiangsu University, Zhenjiang 212013, China. E-mail address: [email protected] (Y. Wang).
https://doi.org/10.1016/j.ijleo.2018.10.095 Received 26 March 2018; Received in revised form 26 September 2018; Accepted 14 October 2018 0030-4026/ © 2018 Elsevier GmbH. All rights reserved.
Optik - International Journal for Light and Electron Optics 178 (2019) 1154–1162
Y. Wang et al.
[20]. In this paper, a phase retrieval method named as matrix p- norm is proposed under the generalized three-step phase-shifting. Among which, the phase shift of each frame can be also extracted easily. Algorithmically, we popularized that the norm p is equal to an even number, and then we made a series of contrast analyses with the change of the value of p. In order to determine the effectiveness of this method, we also discussed its application on red blood cell (RBC). Note that the arbitrary phase shift can be realized by adjusting the angle of the glass plate by an electric rotary table in the link of phase imaging in our experiment, the traditional wave plate or phase shifter is not required. Compared with other algorithms in PSI, this algorithm is more effective and easy to implement. Meanwhile, the only requirement, namely, the fringe limit condition is easy to be met in real cases. 2. Matrix p- norm phase shift and phase retrieval method We define || A ||p as the matrix p- norm. The specific expression is as follows. m
|| A ||p =
1/ p
n
⎛ ⎞ |a |p ⎜∑ ∑ ij ⎟ = = i 1 j 1 ⎝ ⎠
(p is positive integer), (1)
In fact, p is positive integer in this method. || A ||p is a real number, not a plural. In the three- step phase- shifting interferometry, the interference intensity of each frame is
Ikmn = amn + bmn cos[φmn + δk ],
(k = 1, 2, 3)
(2)
In this formula, m and n denote the pixel position of rows and columns of interferograms, respectively. amn , bmn and φmn are the background intensity, the modulation amplitude and the measured phase, respectively. δk is the phase shift related to the kth interferogram. With the measured intensities, the difference between the tth and sth interferograms can be expressed as
ΔIst = Ismn − Itmn = 2bmn sin ⎡φmn + ⎣
δs + δt ⎤ δ − δt ⎤ sin ⎡ s , 2 ⎦ ⎣ 2 ⎦
(s, t = 1, 2, 3)
(3)
Then, according to the above definition, the matrix p- norm value of the intensity difference is
{
M
(
N
⎧ ⎪ 2 sin = ⎨ ⎪ 2 sin ⎩
) {∑ ( ) {∑
(
M N p ∑n = 1 bmn m=1
δs − δt 2
M N p ∑n = 1 bmn m=1
δs − δt 2
(
sinp ϕmn +
(
sinp ϕmn +
) sin (
δs + δt 2
Estp = || ΔIst ||p = ∑m = 1 ∑n = 1 ⎡2bmn sin ϕmn + ⎣
δs + δt 2 δs + δt 2
1
)}
1
)}
p
p
δs − δt 2
p
) ⎤⎦
1
}
p
(p is an even integer)
(p is an odd integer)
(4)
Assuming that the number of interference fringes is more than 1, so the phase change in the observation area is more than 2π (rad) , thus the following approximation condition is satisfied when p is an even number M
N
p sinp ⎡φmn + ∑ ∑ bmn m=1 n=1
⎣
δs + δt ⎤ = 2 ⎦
M
N
p sinp φmn ∑ ∑ bmn
(5)
m=1 n=1
Then, the Eq. (4) can be simplified as M
Estp ≈ 2 sin ⎛ ⎝
δs − δt ⎞ ⎧ ∑ 2 ⎠ ⎨ m=1 ⎩
N
1
⎫ p sinp (ϕmn ) ∑ bmn ⎬
n=1
p
(p is an even integer) (6)
⎭ M
N
p sinp (φmn ) } In order to facilitate the description and simplify the calculation, we define : C p = 2 {∑m = 1 ∑n = 1 bmn Thus in three-step phase shifting interferometry, there are
1 p
and δ1 = 0 .
( ) ( )
⎧ E12p = C p⋅ sin δ2 2 ⎪ ⎪ δ3 p E13p = C ⋅ sin 2 (p is an even integer) ⎨ ⎪ − δ δ 3 2 p ⎪ E23p = C ⋅ sin 2 ⎩
(
)
(7)
In order to avoid the uncertainty of phase shift, we require that the values of δ2 and δ3 are limited to the range [0, π ], thus
| sin(δ2 2)| = sin(δ2 2) , | sin(δ3 2)| = sin(δ3 2) . In addition, by comparing E12p with E13p , we can make out the symbol of sin
(
δ3 − δ2 2
),
that is E13p > E12p , so sin(δ3 2) > sin(δ2 2) and δ3 2 > δ2 2 , therefore the formula | sin[(δ3 − δ2) 2]| = sin[(δ3 − δ2) 2] is established. In this case, the Eq. (7) can be simplified as 1155
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Y. Wang et al.
( ) ( )
⎧ E12p = C p⋅sin δ22 ⎪ ⎪ δ E13p = C p⋅sin 23 (p is an even integer) ⎨ ⎪ δ3 − δ2 ⎪ E23p = C p⋅sin 2 ⎩
(
This parameter
Cp
)
(8)
can be determined by the following type
E13p = E12p ×
E23p 2 1 − ⎛ p ⎞ + E23p × ⎝ C ⎠ ⎜
⎟
E12p 2 1−⎛ p ⎞ ⎝ C ⎠ ⎜
⎟
(9)
After C p
was determined, we substitute into expression E12p and E13p of Eq.(8), to work out δ2 and δ3 . In this way, the whole phase shift acquisition is completed. Then you can calculate the required phase information.
φ = arctan
I3 − I2 + (I1 − I3)cos δ2 + (I2 − I1)cos δ3 (I1 − I3)sin δ2 + (I2 − I1)sin δ3
(10)
3. Numerical simulation and discussion 3.1. Simulation without sample In order to verify the feasibility of the method of phase shift based on the matrix p- norm, we have carried out a series of simulations of a phase plane wave. In the simulation, we assume that the background intensity and the modulation amplitude are set as: amn = 120 × exp[−(m2 + n2)/5002], bmn = 100 × exp[−(m2 + n2)/5002]. The under test phase is φmn = π × Nf × n/500, (Nf = 2) . Among them, m, n for the pixel coordinates, the specific values are: m, n= -500, -499, …, 499, 500, Nf is the fringe number in the fringe pattern. In addition, we assume that the first-, second-, and third-step phase shift are respectively δ1 = 0, δ2 = π /4, δ3 = 3π /4 .Taking into account the existence of the noise in the actual optical experiment, we introduce the signal-tonoise ratio (SNR) of 10 dB the gaussian noise. According to the mentioned parameters as well as the Eq. (9), we can simulate the three phase-shifted interferograms with the size of 1000 × 1000 pixels, as shown in Fig. 1(a–c), we can see the stripes are straight. After the phase shift of each step is determined, we can calculate the phase according to Eq. (10). Fig. 1(d) is the reconstructed wrapped phase of the plane wave. It is wrapped in [−π , π ]. Fig. 1(e) is the unwrapped phase image by the least squares cosine transform. We can be seen that it is a plane wave form. Fig. 1(f) is the theoretical phase. By observing the reconstruction (Fig. 1(e)) and theoretical phases (Fig. 1(f)), we could see that they are no significant difference. Fig. 1(g–i) are the corresponding results when p = 4 , and Fig. 1(j–l) are the results when p = 6 . In order to discuss the accuracy of the reconstructed phase with the method presented above, we calculated the difference between the theoretical phase and the reconstructed one. Fig. 2(a) is the difference between them when p = 2, and Fig. 2(b) is the transverse profiles corresponding to Fig. 2(a). Fig. 2(c–d) are the results when p = 4 , Fig. 2(e–f) are the results when p = 6 . It could be found that most of the absolute values are less than 0.06 rad in Fig. 2(b, d, f). The associated results are listed in Table 1. The p numbers are listed in the first column, the fringe numbers are listed in the second column, the calculated values of δ2 and δ3 are listed in third and fourth columns, respectively, and the associated errors are also given in the corresponding columns. The root-meansquare error (RMSE) and the computation times are listed in fifth and final columns, respectively. From Table1, it could be found that with the increase of the p number, the calculated value of δ2 and δ3 decreases, the absolute values of their corresponding errors both decrease. It indicates that the larger p number, the calculated value of the phase shift is more close to its theoretical value. In addition, the calculated value of δ2 and δ3 are more and more accurate even then time consuming increase, and then it also could be found that the calculated value of δ2 and δ3 , the absolute values of their corresponding errors and the RMSE both decrease with the increase of the fringe number. It indicates that the larger fringe number, the calculated value of the phase shift is more close to its theoretical value too. In order to make the simulation close to the experimental situation, we added noise to the simulation and discuss the algorithm accuracy under different noise intensities in different p number. Different intensities of Gaussian noise which SNR from 10 dB to 70 dB are added to the simulation. For a better comparison analysis, we aggregated and fitted the data into a graph. The associated results are shown in the graph. From the graph, it could be found that with the increase of the noise intensities (SNR is inversely proportional to the noise amplitude), the RMSE is increasing. Simultaneously, we also could find that with the increase of the p number, the RMSE is decreasing under the same noise intensity. This is in line with the simulation results above. 3.2. Simulation with RBC The shape of a human RBC is double concave disc with thin center and thick margin. The diameter of normal RBCs are about 7 ∼ 8.5μm ,the center is about 1.0μm and the margin is about 2.4μm . Mature RBCs do not contain nuclei and organelles. The optical properties mainly determined by the lipid bilayer of cell membrane which has a refractive index of 1.46. The simulation parameters set as follows. The diameter of RBC model is 7.55μm , the maximum and minimum thickness is 2.54μm and 1.21μm , respectively. The refractive indices of model and the air are 1.40 and 1.34. The other parameters are same like above. As an example with p = 4 . Three 1156
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Fig. 1. The simulated results：(a), (b) and (c) the phase shift interferometry, (d), (g) and (j) the wrapped phase, (e), (h) and (k) the unwrapped phase, (f), (i) and (l) the theoretical phase.
phase shifted interference patterns(512 × 512 ) were simulated, as shown in Fig. 3(a–c). Fig. 3 (d) is 2D phase obtained by least squares method and Fig. 3(e) is 3D phase. Two calculations of δ2 and δ3 is 0.7896 rad and 2.3469 rad, respectively. Their corresponding deviations are 0.0046 rad and -0.0081 rad, respectively. It could be seen that the calculation error of phase shift is very small, and it shows the correctness and accuracy of the method again. After each phase shift is determined, the phase can be calculated according to the formula. In the next, we also calculated the difference between the theoretical phase and the reconstructed one. Fig. 4(a) is theoretical phase. Fig. 4(b) is the difference between them, and Fig. 4(c) is the transverse profiles corresponding to Fig. 4(b). It could be found that most of the values fluctuate within a range of -0.02 rad to 0.02 rad, with a maximum of 0.04 rad. Moreover, the result of RMSE is 0.0192 rad. In terms of accuracy, the reconstruction precision of this method is very high, and the similarity between reconstructed and theoretical phases is above 0.99. In terms of run time, we take p = 4 for example in order to discuss the efficiency. Via the measurement, the running time of the proposed method is 0.4376 s, that of Hilbert method is 0.6703 s. Operational efficiency was increased by 34.7%. 4. Experimental verification 4.1. Algorithm verification In order to further verify the feasibility of the three step generalized phase shift interferometry phase imaging method, we have 1157
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Fig. 2. (a) the difference between Fig. 1(e) and (f), (c) the difference between Fig. 1 (h) and (i), (e) the difference between Fig. 1 (k) and (l), (b), (d) and (f) the transverse profiles corresponding to Fig. 2(a), (c) and (e), respectively. Table 1 Phase shift extraction in different p number and different fringe number. p
Nf
Calculated δ2 and its error (rad)
Calculated δ3 and its error(rad)
RMSE (rad)
Times (rad)
2
1 2 3 4 5 1 2 3 4 5 1 2 3 4 5
0.7933 0.7894 0.7873 0.7865 0.7862 0.7472 0.7863 0.7859 0.7858 0.7857 0.7171 0.7859 0.7856 0.7856 0.7856
2.3641 2.3603 2.3582 2.3574 2.3571 2.3148 2.3570 2.3568 2.3567 2.3565 2.2755 2.3568 2.3566 2.3566 2.3565
0.0204 0.0181 0.0162 0.0148 0.0138 0.0241 0.0204 0.0181 0.0164 0.0134 0.0377 0.0205 0.0184 0.0161 0.0141
0.2970 0.3102 0.3054 0.3099 0.3033 0.4020 0.3915 0.4055 0.3866 0.3913 0.4508 0.4478 0.4308 0.4687 0.4480
4
6
0.0083 0.0044 0.0023 0.0015 0.0012 −0.0378 0.0013 0.0009 0.0008 0.0007 −0.0679 0.0009 0.0006 0.0006 0.0006
0.0091 0.0053 0.0032 0.0024 0.0021 −0.0402 0.0020 0.0018 0.0017 0.0015 −0.0795 0.0018 0.0016 0.0016 0.0015
carried out the following optical experiments. Experimental optical path is shown in Fig. 5, the principle is based on Mach - Zehnder optical interference principle. The illumination source is a He-Ne laser with the wavelength of 632.8 nm, and the phase-shift controler is a electric rotating platform (Sigma SGSP-40YAW) with the angle of 0.005 each step. The beam of the laser is first passed through a beam expander system. The beam expander system is composed of two aberration elimination lenses and a pinhole spatial filter. It continued to move forward through the beam splitter BS1, divided into the sample light and the reference light. Then they bunched by the beam splitter BS2, the interference fringes are formed on the CMOS. The interferogram is captured by the CMOS with size of 1000 × 1000 pixels (4 mm × 4 mm). In order to realize the three-step phase shift interferometry, we need to carry out the phase1158
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Fig. 3. The simulated results of RBC：(a), (b) and (c) the phase shift interferometry, (d) the unwrapped phase, (e) the 3D phase.
Fig. 4. (a) the theoretical phase, (b) the difference between Figs. 3(d) and 4 (a), (c) the transverse profiles corresponding to Fig. 4(b).
Fig. 5. Experimental setup for phase-shifting interferometry. BE: beam expander, BS: beam splitter, M: mirror, L: lens, MO: microscopy objective.
shifting of the reference light twice times. It should be noted that we placed a composed of electric rotary platform and a glass plate of the phase shifter in the reference arm, and the glass plate was fixed on the electric rotary platform in a vertical mesa manner, the electric rotary platform was connected with the controller and the computer to control the glass plate for precise rotation, so as to achieve phase shifting. We obtained three interference patterns when θ1 = 0∘, θ2 = 2∘, θ3 = 2.45∘. According to the relationship between the phase shift δ and the tilt angle θ of the glass plate, that is, δ ≈ πd (n − 1) θ 2/(nλ ) , the wavelength of the light source is λ , and the thickness and refractive index of the glass plate are d and n, respectively. In our experiment, λ = 632.8nm , d = 1mm , n = 1.5. Obviously, the phase shifts of 1th, 2th and 3th frames are respectively 0 rad, 2 rad and 3 rad theoretically. In order to generate the straight interference fringes corresponding to the plane wave, we put two of the same lens on the reference arm and the sample arm of the experimental setup (Fig. 5), respectively, and the focal length is 50 mm. By adjusting the angle of the mirror M1, the reference field is tilted slightly to the sample field, thereby producing a homogeneous off-axis interference fringe. According to the above method, we can adjust the phase shifter to collect three different phase shift of the straight interference 1159
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Fig. 6. The experimental results：(a), (b) and (c) the phase shift interferometry (d), (f) and (h) the wrapped phase (e), (g) and (i) the unwrapped phase.
fringes, as shown in Fig. 6(a–c). According to the method proposed in this paper, by using the three images, we can obtain the phase shift, and realize the phase reconstruction. Fig. 6(d) is the reproduced wrapped phase. Fig. 6(e) is the 2D display of phase distribution. Fig. 6(f–g) are the reproduced wrapped phase and the 2D display of phase distribution when p = 4 . Fig. 4(h–i) are the reproduced wrapped phase and the 2D display of phase distribution when p = 6. The p numbers are taken as an example and the associated results are listed in Table 2. The configuration is the same as those of the above table. From Table 2, it also could be found that with the increase of the p number, the calculated value of δ2 and δ3 decreases, the absolute values of their corresponding errors both decrease. In addition, the calculated value of δ2 and δ3 are more and more accurate even then time consuming increase too. We can see the method is highly effective and accurate, well suited for a broad (blind taken) phase shifting interferometry. 4.2. Experimental verification with RBC Firstly, the blood samples were centrifuged and extracted to obtain RBCs, and then diluted and placed in a saline environment with a refractive index of about 1.33. After the sample preparation, it could be put into the imaging system for experiment. Three phase shift interference patterns are shown in Fig. 7(a–c). Via the calculation, δ2 and δ3 are 2. 3980 rad and 2.7142 rad, respectively. The corresponding deviations of their theoretical value are 0.3980 rad and -0.2858 rad, respectively. This error is slightly larger than simulation, but it retains within the acceptance range due to the disturbance and influence of the experimental environment. Finally, the RBC phase was obtained as shown in Fig. 7(d). The morphological profile of RBC can be clearly seen from it. 5. Conclusions The matrix p- norm phase retrieval method under the generalized three-step phase-shifting is proposed in the paper and then Table 2 Phase shift extraction in different p number. p
Calculated δ2 and its error (rad)
2 4 6
2.1152 2.0927 2.0049
Calculated δ3 and its error(rad) 0.1152 0.0927 0.0049
3.1157 3.1033 2.9913
1160
Time(s)
0.1157 0.1033 −0.0087
0.3207 1.0325 1.3509
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Fig. 7. The experimental results of RBC：(a), (b) and (c) the phase shift interferometry, (d) the unwrapped phase.
Graph 1. The RMSE under different noise intensity.
applied on a RBC. Although in this method, the fringe number should greater than 1, but compared to other conventional phase retrieval methods, we don’t need to eliminate background light intensity or obtain the beam intensity. The associated algorithm is applied to the simulated and experimental interference patterns. Its efficiency was increased by thirty percent compared with Hilbert method, and the reducibility of RBC in the simulation is similar to it. The experimental results also demonstrate its correctness in spite of the error caused by environmental disturbance. These results show that this method is suitable for blind PSI with high accuracy (Graph 1 ). Acknowledgement This work was supported by National Natural Science Foundation of China (Nos. 11604127, 11874184 and 11474134), Postdoctoral Science Fund of Jiangsu Province (No. 2018K032A), Natural Science Foundation of Jiangsu Province (No. SBK2015042429), Foundation of Six Kinds of High Talented People of Jiangsu Province (No. 2015-DZXX-023) and Graduate Student Scientific Research Innovation Projects in Jiangsu Province (No: KYCX18_2226). References [1] L.Z. Cai, Q. Liu, X.L. Yang, Phase-shift extraction and wave-front reconstruction in phase-shifting interferometry with arbitrary phase steps, Opt. Lett. 28 (19) (2003) 1808–1810. [2] L.Z. Cai, Q. Liu, X.L. Yang, Generalized phase-shifting interferometry with arbitrary unknown phase steps for diffraction objects, Opt. Lett. 29 (2) (2004) 183–185. [3] Y.Y. Xu, Y.W. Wang, et al., Three-frame generalized phase-shifting interferometry by a Euclidean matrix norm algorithm, Opt. Lasers Eng. 84 (2016) 89–95. [4] Y.Y. Xu, Y.W. Wang, et al., A derivative method of phase retrieval based on two interferograms with an unknown phase shift, Optik 127 (1) (2016) 326–330. [5] I. Yamaguchi, Principles and applications of phase-shifting digital holography, Opt. Lett. 22 (16) (1997) 1268–1270. [6] I. Yamaguchi, J. Kato, et al., Image formation in phase-shifting digital holography and applications to microscopy, Appl. Opt. 40 (34) (2001) 6177–6186. [7] N. Yoshikawa, Phase determination method in statistical generalized phase-shifting digital holography, Appl. Opt. 52 (52) (2013) 1947–1953. [8] N. Yoshikawa, T. Shiratori, et al., Robust phase-shift estimation method for statistical generalized phase-shifting digital holography, Opt. Express 22 (12) (2014) 14155–14165. [9] X.E. Xu, L.Z. Cai, et al., Simple direct extraction of unknown phase shift and wavefront reconstruction in generalized phase-shifting interferometry: algorithm and experiments, Opt. Lett. 33 (8) (2008) 776–778. [10] X.F. Xu, L.Z. Cai, et al., Direct phase shift extraction and wavefront reconstruction in two-step generalized phase-shifting interferometry, J. Opt. 12 (1) (2010) 74–77. [11] X.F. Xu, L.Z. Cai, et al., Accurate phase shift extraction for generalized phase-shifting interferometry, Chin. Phys. Lett. 27 (2) (2010) 024215. [12] J.C. Xu, Y. Li, et al., Phase-shift extraction for phase-shifting interferometry by histogram of phase difference, Opt. Express 18 (23) (2010) 24368–24378. [13] X. Chen, M. Gramaglia, et al., Phase-shifting interferometry with uncalibrated phase shifts, Appl. Opt. 39 (4) (2000) 584–591. [14] C.S. Guo, B. Sha, et al., Zero difference algorithm for phase shift extraction in blind phase-shifting holography, Opt. Lett. 39 (4) (2014) 813–816. [15] L.H. Fei, X.X. Lu, et al., Single-wavelength phase retrieval method from simultaneous multi-wavelength in-line phase-shifting interferograms, Opt. Express 22 (25) (2014) 30910–30923. [16] Z. Wang, B. Han, et al., Advanced iterative algorithm for phase extraction of randomly phase-shifted interferograms, Opt. Lett. 29 (14) (2004) 1671–1673. [17] X.F. Xu, L.Z. Cai, et al., Generalized phase-shifting interferometry with arbitrary unknown phase shifts: direct wave-front reconstruction by blind phase shift extraction and its experimental verification, Appl. Phys. Lett. 90 (12) (2007) 121–124. [18] X.F. Xu, L.Z. Cai, et al., Simple direct extraction of unknown phase shift and wavefront reconstruction in generalized phase-shifting interferometry: algorithm and experiments, Opt. Lett. 33 (8) (2008) 776–778. [19] J.C. Xu, Y. Li, et al., Phase-shift extraction for phase-shifting interferometry by histogram of phase difference, Opt. Express 18 (23) (2010) 24368–24378.
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Original research article
Matrix p- norm for phase retrieval under generalized three-step phase-shifting and its application on a red blood cell
T
⁎
Yawei Wanga,b, , Hao Hanb, Yuanyuan Xua, Ying Jia a b
Faculty of Science, Engineering, Jiangsu University, Zhenjiang, 212013, China School of Mechanical Engineering, Jiangsu University, Zhenjiang, 212013, China
A R T IC LE I N F O
ABS TRA CT
Keywords: Generalized phase-shifting interferometry Matrix p- norm Red blood cell Phase retrieval
Phase-shifting interferometry (PSI) is an important technique, which has been widely used in optical measurement and microscopy. Based on the matrix p- norm algorithm, a phase retrieval method applied to the generalized three-step PSI is proposed in this paper. In this method, only three phase-shifted interferograms are required without anyother measurements. When the norm p is an even number, we can obtain the good results. As for the experiment, the setup is simple in structure and easy for experimental operation. The result of a plane wavefront suggestes its accuracy. Both simulation and experimental results of a red blood cell verify that the effectiveness and practicability of this method.
1. Introduction As an important technical mean for optical measurement and microscopy, phase-shifting interferometry (PSI) has widely used [1–14]. In traditional PSI, a special and constant phase shift in each frame or the equal phase shift between two adjacent frames is often required. However, the actual value of the phase shift is slightly different from its theoretical value due to some errors in practice. In order to settle such a serious problem, many methods have been reported [15–23], among which, the generalized phase shift exaction techniques provide a possibility for extracting arbitrary unknown phase shifts from holograms directly. For example, the research group of Fei [15] applied the iterative algorithm to extract the phase shift of each single-wavelength from simultaneous phase-shifting multi-wavelength interferograms. Wang [16] proposed an advanced iterative algorithm based on the least squares method to determine the phase shift and the phase distribution simultaneously. Comparing with the methods mentioned above, the following ones have attracted more attention for the less computational loads and rapid processing rate. Cai and other researchers proposed a class of statistical algorithms to calculate the unknown phase shift directly and reconstruct the wavefront from two or more holograms [17,18]. In these algorithms, the statistical property of the diffraction field of the measured object is required to be considered. Besides, some other measurements, such as the intensity of reference wave, are also needed. Arming at the limitation of the statistical property, Xu [19] proposed an advanced method to extract the phase shift accurately, according to the histogram of phase difference between two adjacent frames. However, it is required to determine both the background intensity and the modulation amplitude by searching for the maximum and the minimum intensity in each pixel of interferograms. As a result, this method is very time-consuming. And then an asynchronous phase-shifting method based on the principal component analysis (PCA) is proposed with high precision and rapid speed, but it is required to eliminate the background term by a temporal average in advance
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Corresponding author at: Faculty of Science, Jiangsu University, Zhenjiang 212013, China. E-mail address: [email protected] (Y. Wang).
https://doi.org/10.1016/j.ijleo.2018.10.095 Received 26 March 2018; Received in revised form 26 September 2018; Accepted 14 October 2018 0030-4026/ © 2018 Elsevier GmbH. All rights reserved.
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[20]. In this paper, a phase retrieval method named as matrix p- norm is proposed under the generalized three-step phase-shifting. Among which, the phase shift of each frame can be also extracted easily. Algorithmically, we popularized that the norm p is equal to an even number, and then we made a series of contrast analyses with the change of the value of p. In order to determine the effectiveness of this method, we also discussed its application on red blood cell (RBC). Note that the arbitrary phase shift can be realized by adjusting the angle of the glass plate by an electric rotary table in the link of phase imaging in our experiment, the traditional wave plate or phase shifter is not required. Compared with other algorithms in PSI, this algorithm is more effective and easy to implement. Meanwhile, the only requirement, namely, the fringe limit condition is easy to be met in real cases. 2. Matrix p- norm phase shift and phase retrieval method We define || A ||p as the matrix p- norm. The specific expression is as follows. m
|| A ||p =
1/ p
n
⎛ ⎞ |a |p ⎜∑ ∑ ij ⎟ = = i 1 j 1 ⎝ ⎠
(p is positive integer), (1)
In fact, p is positive integer in this method. || A ||p is a real number, not a plural. In the three- step phase- shifting interferometry, the interference intensity of each frame is
Ikmn = amn + bmn cos[φmn + δk ],
(k = 1, 2, 3)
(2)
In this formula, m and n denote the pixel position of rows and columns of interferograms, respectively. amn , bmn and φmn are the background intensity, the modulation amplitude and the measured phase, respectively. δk is the phase shift related to the kth interferogram. With the measured intensities, the difference between the tth and sth interferograms can be expressed as
ΔIst = Ismn − Itmn = 2bmn sin ⎡φmn + ⎣
δs + δt ⎤ δ − δt ⎤ sin ⎡ s , 2 ⎦ ⎣ 2 ⎦
(s, t = 1, 2, 3)
(3)
Then, according to the above definition, the matrix p- norm value of the intensity difference is
{
M
(
N
⎧ ⎪ 2 sin = ⎨ ⎪ 2 sin ⎩
) {∑ ( ) {∑
(
M N p ∑n = 1 bmn m=1
δs − δt 2
M N p ∑n = 1 bmn m=1
δs − δt 2
(
sinp ϕmn +
(
sinp ϕmn +
) sin (
δs + δt 2
Estp = || ΔIst ||p = ∑m = 1 ∑n = 1 ⎡2bmn sin ϕmn + ⎣
δs + δt 2 δs + δt 2
1
)}
1
)}
p
p
δs − δt 2
p
) ⎤⎦
1
}
p
(p is an even integer)
(p is an odd integer)
(4)
Assuming that the number of interference fringes is more than 1, so the phase change in the observation area is more than 2π (rad) , thus the following approximation condition is satisfied when p is an even number M
N
p sinp ⎡φmn + ∑ ∑ bmn m=1 n=1
⎣
δs + δt ⎤ = 2 ⎦
M
N
p sinp φmn ∑ ∑ bmn
(5)
m=1 n=1
Then, the Eq. (4) can be simplified as M
Estp ≈ 2 sin ⎛ ⎝
δs − δt ⎞ ⎧ ∑ 2 ⎠ ⎨ m=1 ⎩
N
1
⎫ p sinp (ϕmn ) ∑ bmn ⎬
n=1
p
(p is an even integer) (6)
⎭ M
N
p sinp (φmn ) } In order to facilitate the description and simplify the calculation, we define : C p = 2 {∑m = 1 ∑n = 1 bmn Thus in three-step phase shifting interferometry, there are
1 p
and δ1 = 0 .
( ) ( )
⎧ E12p = C p⋅ sin δ2 2 ⎪ ⎪ δ3 p E13p = C ⋅ sin 2 (p is an even integer) ⎨ ⎪ − δ δ 3 2 p ⎪ E23p = C ⋅ sin 2 ⎩
(
)
(7)
In order to avoid the uncertainty of phase shift, we require that the values of δ2 and δ3 are limited to the range [0, π ], thus
| sin(δ2 2)| = sin(δ2 2) , | sin(δ3 2)| = sin(δ3 2) . In addition, by comparing E12p with E13p , we can make out the symbol of sin
(
δ3 − δ2 2
),
that is E13p > E12p , so sin(δ3 2) > sin(δ2 2) and δ3 2 > δ2 2 , therefore the formula | sin[(δ3 − δ2) 2]| = sin[(δ3 − δ2) 2] is established. In this case, the Eq. (7) can be simplified as 1155
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( ) ( )
⎧ E12p = C p⋅sin δ22 ⎪ ⎪ δ E13p = C p⋅sin 23 (p is an even integer) ⎨ ⎪ δ3 − δ2 ⎪ E23p = C p⋅sin 2 ⎩
(
This parameter
Cp
)
(8)
can be determined by the following type
E13p = E12p ×
E23p 2 1 − ⎛ p ⎞ + E23p × ⎝ C ⎠ ⎜
⎟
E12p 2 1−⎛ p ⎞ ⎝ C ⎠ ⎜
⎟
(9)
After C p
was determined, we substitute into expression E12p and E13p of Eq.(8), to work out δ2 and δ3 . In this way, the whole phase shift acquisition is completed. Then you can calculate the required phase information.
φ = arctan
I3 − I2 + (I1 − I3)cos δ2 + (I2 − I1)cos δ3 (I1 − I3)sin δ2 + (I2 − I1)sin δ3
(10)
3. Numerical simulation and discussion 3.1. Simulation without sample In order to verify the feasibility of the method of phase shift based on the matrix p- norm, we have carried out a series of simulations of a phase plane wave. In the simulation, we assume that the background intensity and the modulation amplitude are set as: amn = 120 × exp[−(m2 + n2)/5002], bmn = 100 × exp[−(m2 + n2)/5002]. The under test phase is φmn = π × Nf × n/500, (Nf = 2) . Among them, m, n for the pixel coordinates, the specific values are: m, n= -500, -499, …, 499, 500, Nf is the fringe number in the fringe pattern. In addition, we assume that the first-, second-, and third-step phase shift are respectively δ1 = 0, δ2 = π /4, δ3 = 3π /4 .Taking into account the existence of the noise in the actual optical experiment, we introduce the signal-tonoise ratio (SNR) of 10 dB the gaussian noise. According to the mentioned parameters as well as the Eq. (9), we can simulate the three phase-shifted interferograms with the size of 1000 × 1000 pixels, as shown in Fig. 1(a–c), we can see the stripes are straight. After the phase shift of each step is determined, we can calculate the phase according to Eq. (10). Fig. 1(d) is the reconstructed wrapped phase of the plane wave. It is wrapped in [−π , π ]. Fig. 1(e) is the unwrapped phase image by the least squares cosine transform. We can be seen that it is a plane wave form. Fig. 1(f) is the theoretical phase. By observing the reconstruction (Fig. 1(e)) and theoretical phases (Fig. 1(f)), we could see that they are no significant difference. Fig. 1(g–i) are the corresponding results when p = 4 , and Fig. 1(j–l) are the results when p = 6 . In order to discuss the accuracy of the reconstructed phase with the method presented above, we calculated the difference between the theoretical phase and the reconstructed one. Fig. 2(a) is the difference between them when p = 2, and Fig. 2(b) is the transverse profiles corresponding to Fig. 2(a). Fig. 2(c–d) are the results when p = 4 , Fig. 2(e–f) are the results when p = 6 . It could be found that most of the absolute values are less than 0.06 rad in Fig. 2(b, d, f). The associated results are listed in Table 1. The p numbers are listed in the first column, the fringe numbers are listed in the second column, the calculated values of δ2 and δ3 are listed in third and fourth columns, respectively, and the associated errors are also given in the corresponding columns. The root-meansquare error (RMSE) and the computation times are listed in fifth and final columns, respectively. From Table1, it could be found that with the increase of the p number, the calculated value of δ2 and δ3 decreases, the absolute values of their corresponding errors both decrease. It indicates that the larger p number, the calculated value of the phase shift is more close to its theoretical value. In addition, the calculated value of δ2 and δ3 are more and more accurate even then time consuming increase, and then it also could be found that the calculated value of δ2 and δ3 , the absolute values of their corresponding errors and the RMSE both decrease with the increase of the fringe number. It indicates that the larger fringe number, the calculated value of the phase shift is more close to its theoretical value too. In order to make the simulation close to the experimental situation, we added noise to the simulation and discuss the algorithm accuracy under different noise intensities in different p number. Different intensities of Gaussian noise which SNR from 10 dB to 70 dB are added to the simulation. For a better comparison analysis, we aggregated and fitted the data into a graph. The associated results are shown in the graph. From the graph, it could be found that with the increase of the noise intensities (SNR is inversely proportional to the noise amplitude), the RMSE is increasing. Simultaneously, we also could find that with the increase of the p number, the RMSE is decreasing under the same noise intensity. This is in line with the simulation results above. 3.2. Simulation with RBC The shape of a human RBC is double concave disc with thin center and thick margin. The diameter of normal RBCs are about 7 ∼ 8.5μm ,the center is about 1.0μm and the margin is about 2.4μm . Mature RBCs do not contain nuclei and organelles. The optical properties mainly determined by the lipid bilayer of cell membrane which has a refractive index of 1.46. The simulation parameters set as follows. The diameter of RBC model is 7.55μm , the maximum and minimum thickness is 2.54μm and 1.21μm , respectively. The refractive indices of model and the air are 1.40 and 1.34. The other parameters are same like above. As an example with p = 4 . Three 1156
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Fig. 1. The simulated results：(a), (b) and (c) the phase shift interferometry, (d), (g) and (j) the wrapped phase, (e), (h) and (k) the unwrapped phase, (f), (i) and (l) the theoretical phase.
phase shifted interference patterns(512 × 512 ) were simulated, as shown in Fig. 3(a–c). Fig. 3 (d) is 2D phase obtained by least squares method and Fig. 3(e) is 3D phase. Two calculations of δ2 and δ3 is 0.7896 rad and 2.3469 rad, respectively. Their corresponding deviations are 0.0046 rad and -0.0081 rad, respectively. It could be seen that the calculation error of phase shift is very small, and it shows the correctness and accuracy of the method again. After each phase shift is determined, the phase can be calculated according to the formula. In the next, we also calculated the difference between the theoretical phase and the reconstructed one. Fig. 4(a) is theoretical phase. Fig. 4(b) is the difference between them, and Fig. 4(c) is the transverse profiles corresponding to Fig. 4(b). It could be found that most of the values fluctuate within a range of -0.02 rad to 0.02 rad, with a maximum of 0.04 rad. Moreover, the result of RMSE is 0.0192 rad. In terms of accuracy, the reconstruction precision of this method is very high, and the similarity between reconstructed and theoretical phases is above 0.99. In terms of run time, we take p = 4 for example in order to discuss the efficiency. Via the measurement, the running time of the proposed method is 0.4376 s, that of Hilbert method is 0.6703 s. Operational efficiency was increased by 34.7%. 4. Experimental verification 4.1. Algorithm verification In order to further verify the feasibility of the three step generalized phase shift interferometry phase imaging method, we have 1157
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Fig. 2. (a) the difference between Fig. 1(e) and (f), (c) the difference between Fig. 1 (h) and (i), (e) the difference between Fig. 1 (k) and (l), (b), (d) and (f) the transverse profiles corresponding to Fig. 2(a), (c) and (e), respectively. Table 1 Phase shift extraction in different p number and different fringe number. p
Nf
Calculated δ2 and its error (rad)
Calculated δ3 and its error(rad)
RMSE (rad)
Times (rad)
2
1 2 3 4 5 1 2 3 4 5 1 2 3 4 5
0.7933 0.7894 0.7873 0.7865 0.7862 0.7472 0.7863 0.7859 0.7858 0.7857 0.7171 0.7859 0.7856 0.7856 0.7856
2.3641 2.3603 2.3582 2.3574 2.3571 2.3148 2.3570 2.3568 2.3567 2.3565 2.2755 2.3568 2.3566 2.3566 2.3565
0.0204 0.0181 0.0162 0.0148 0.0138 0.0241 0.0204 0.0181 0.0164 0.0134 0.0377 0.0205 0.0184 0.0161 0.0141
0.2970 0.3102 0.3054 0.3099 0.3033 0.4020 0.3915 0.4055 0.3866 0.3913 0.4508 0.4478 0.4308 0.4687 0.4480
4
6
0.0083 0.0044 0.0023 0.0015 0.0012 −0.0378 0.0013 0.0009 0.0008 0.0007 −0.0679 0.0009 0.0006 0.0006 0.0006
0.0091 0.0053 0.0032 0.0024 0.0021 −0.0402 0.0020 0.0018 0.0017 0.0015 −0.0795 0.0018 0.0016 0.0016 0.0015
carried out the following optical experiments. Experimental optical path is shown in Fig. 5, the principle is based on Mach - Zehnder optical interference principle. The illumination source is a He-Ne laser with the wavelength of 632.8 nm, and the phase-shift controler is a electric rotating platform (Sigma SGSP-40YAW) with the angle of 0.005 each step. The beam of the laser is first passed through a beam expander system. The beam expander system is composed of two aberration elimination lenses and a pinhole spatial filter. It continued to move forward through the beam splitter BS1, divided into the sample light and the reference light. Then they bunched by the beam splitter BS2, the interference fringes are formed on the CMOS. The interferogram is captured by the CMOS with size of 1000 × 1000 pixels (4 mm × 4 mm). In order to realize the three-step phase shift interferometry, we need to carry out the phase1158
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Fig. 3. The simulated results of RBC：(a), (b) and (c) the phase shift interferometry, (d) the unwrapped phase, (e) the 3D phase.
Fig. 4. (a) the theoretical phase, (b) the difference between Figs. 3(d) and 4 (a), (c) the transverse profiles corresponding to Fig. 4(b).
Fig. 5. Experimental setup for phase-shifting interferometry. BE: beam expander, BS: beam splitter, M: mirror, L: lens, MO: microscopy objective.
shifting of the reference light twice times. It should be noted that we placed a composed of electric rotary platform and a glass plate of the phase shifter in the reference arm, and the glass plate was fixed on the electric rotary platform in a vertical mesa manner, the electric rotary platform was connected with the controller and the computer to control the glass plate for precise rotation, so as to achieve phase shifting. We obtained three interference patterns when θ1 = 0∘, θ2 = 2∘, θ3 = 2.45∘. According to the relationship between the phase shift δ and the tilt angle θ of the glass plate, that is, δ ≈ πd (n − 1) θ 2/(nλ ) , the wavelength of the light source is λ , and the thickness and refractive index of the glass plate are d and n, respectively. In our experiment, λ = 632.8nm , d = 1mm , n = 1.5. Obviously, the phase shifts of 1th, 2th and 3th frames are respectively 0 rad, 2 rad and 3 rad theoretically. In order to generate the straight interference fringes corresponding to the plane wave, we put two of the same lens on the reference arm and the sample arm of the experimental setup (Fig. 5), respectively, and the focal length is 50 mm. By adjusting the angle of the mirror M1, the reference field is tilted slightly to the sample field, thereby producing a homogeneous off-axis interference fringe. According to the above method, we can adjust the phase shifter to collect three different phase shift of the straight interference 1159
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Fig. 6. The experimental results：(a), (b) and (c) the phase shift interferometry (d), (f) and (h) the wrapped phase (e), (g) and (i) the unwrapped phase.
fringes, as shown in Fig. 6(a–c). According to the method proposed in this paper, by using the three images, we can obtain the phase shift, and realize the phase reconstruction. Fig. 6(d) is the reproduced wrapped phase. Fig. 6(e) is the 2D display of phase distribution. Fig. 6(f–g) are the reproduced wrapped phase and the 2D display of phase distribution when p = 4 . Fig. 4(h–i) are the reproduced wrapped phase and the 2D display of phase distribution when p = 6. The p numbers are taken as an example and the associated results are listed in Table 2. The configuration is the same as those of the above table. From Table 2, it also could be found that with the increase of the p number, the calculated value of δ2 and δ3 decreases, the absolute values of their corresponding errors both decrease. In addition, the calculated value of δ2 and δ3 are more and more accurate even then time consuming increase too. We can see the method is highly effective and accurate, well suited for a broad (blind taken) phase shifting interferometry. 4.2. Experimental verification with RBC Firstly, the blood samples were centrifuged and extracted to obtain RBCs, and then diluted and placed in a saline environment with a refractive index of about 1.33. After the sample preparation, it could be put into the imaging system for experiment. Three phase shift interference patterns are shown in Fig. 7(a–c). Via the calculation, δ2 and δ3 are 2. 3980 rad and 2.7142 rad, respectively. The corresponding deviations of their theoretical value are 0.3980 rad and -0.2858 rad, respectively. This error is slightly larger than simulation, but it retains within the acceptance range due to the disturbance and influence of the experimental environment. Finally, the RBC phase was obtained as shown in Fig. 7(d). The morphological profile of RBC can be clearly seen from it. 5. Conclusions The matrix p- norm phase retrieval method under the generalized three-step phase-shifting is proposed in the paper and then Table 2 Phase shift extraction in different p number. p
Calculated δ2 and its error (rad)
2 4 6
2.1152 2.0927 2.0049
Calculated δ3 and its error(rad) 0.1152 0.0927 0.0049
3.1157 3.1033 2.9913
1160
Time(s)
0.1157 0.1033 −0.0087
0.3207 1.0325 1.3509
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Fig. 7. The experimental results of RBC：(a), (b) and (c) the phase shift interferometry, (d) the unwrapped phase.
Graph 1. The RMSE under different noise intensity.
applied on a RBC. Although in this method, the fringe number should greater than 1, but compared to other conventional phase retrieval methods, we don’t need to eliminate background light intensity or obtain the beam intensity. The associated algorithm is applied to the simulated and experimental interference patterns. Its efficiency was increased by thirty percent compared with Hilbert method, and the reducibility of RBC in the simulation is similar to it. The experimental results also demonstrate its correctness in spite of the error caused by environmental disturbance. These results show that this method is suitable for blind PSI with high accuracy (Graph 1 ). Acknowledgement This work was supported by National Natural Science Foundation of China (Nos. 11604127, 11874184 and 11474134), Postdoctoral Science Fund of Jiangsu Province (No. 2018K032A), Natural Science Foundation of Jiangsu Province (No. SBK2015042429), Foundation of Six Kinds of High Talented People of Jiangsu Province (No. 2015-DZXX-023) and Graduate Student Scientific Research Innovation Projects in Jiangsu Province (No: KYCX18_2226). References [1] L.Z. Cai, Q. Liu, X.L. Yang, Phase-shift extraction and wave-front reconstruction in phase-shifting interferometry with arbitrary phase steps, Opt. Lett. 28 (19) (2003) 1808–1810. [2] L.Z. Cai, Q. Liu, X.L. Yang, Generalized phase-shifting interferometry with arbitrary unknown phase steps for diffraction objects, Opt. Lett. 29 (2) (2004) 183–185. [3] Y.Y. Xu, Y.W. Wang, et al., Three-frame generalized phase-shifting interferometry by a Euclidean matrix norm algorithm, Opt. Lasers Eng. 84 (2016) 89–95. [4] Y.Y. Xu, Y.W. Wang, et al., A derivative method of phase retrieval based on two interferograms with an unknown phase shift, Optik 127 (1) (2016) 326–330. [5] I. Yamaguchi, Principles and applications of phase-shifting digital holography, Opt. Lett. 22 (16) (1997) 1268–1270. [6] I. Yamaguchi, J. Kato, et al., Image formation in phase-shifting digital holography and applications to microscopy, Appl. Opt. 40 (34) (2001) 6177–6186. [7] N. Yoshikawa, Phase determination method in statistical generalized phase-shifting digital holography, Appl. Opt. 52 (52) (2013) 1947–1953. [8] N. Yoshikawa, T. Shiratori, et al., Robust phase-shift estimation method for statistical generalized phase-shifting digital holography, Opt. Express 22 (12) (2014) 14155–14165. [9] X.E. Xu, L.Z. Cai, et al., Simple direct extraction of unknown phase shift and wavefront reconstruction in generalized phase-shifting interferometry: algorithm and experiments, Opt. Lett. 33 (8) (2008) 776–778. [10] X.F. Xu, L.Z. Cai, et al., Direct phase shift extraction and wavefront reconstruction in two-step generalized phase-shifting interferometry, J. Opt. 12 (1) (2010) 74–77. [11] X.F. Xu, L.Z. Cai, et al., Accurate phase shift extraction for generalized phase-shifting interferometry, Chin. Phys. Lett. 27 (2) (2010) 024215. [12] J.C. Xu, Y. Li, et al., Phase-shift extraction for phase-shifting interferometry by histogram of phase difference, Opt. Express 18 (23) (2010) 24368–24378. [13] X. Chen, M. Gramaglia, et al., Phase-shifting interferometry with uncalibrated phase shifts, Appl. Opt. 39 (4) (2000) 584–591. [14] C.S. Guo, B. Sha, et al., Zero difference algorithm for phase shift extraction in blind phase-shifting holography, Opt. Lett. 39 (4) (2014) 813–816. [15] L.H. Fei, X.X. Lu, et al., Single-wavelength phase retrieval method from simultaneous multi-wavelength in-line phase-shifting interferograms, Opt. Express 22 (25) (2014) 30910–30923. [16] Z. Wang, B. Han, et al., Advanced iterative algorithm for phase extraction of randomly phase-shifted interferograms, Opt. Lett. 29 (14) (2004) 1671–1673. [17] X.F. Xu, L.Z. Cai, et al., Generalized phase-shifting interferometry with arbitrary unknown phase shifts: direct wave-front reconstruction by blind phase shift extraction and its experimental verification, Appl. Phys. Lett. 90 (12) (2007) 121–124. [18] X.F. Xu, L.Z. Cai, et al., Simple direct extraction of unknown phase shift and wavefront reconstruction in generalized phase-shifting interferometry: algorithm and experiments, Opt. Lett. 33 (8) (2008) 776–778. [19] J.C. Xu, Y. Li, et al., Phase-shift extraction for phase-shifting interferometry by histogram of phase difference, Opt. Express 18 (23) (2010) 24368–24378.
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