Phase retrieval approach based on the normalized difference maps induced by three interferograms with unknown phase shifts

Hanlin Wang; Chunshu Luo; Liyun Zhong; Shuzhen Ma; Xiaoxu Lu

doi:10.1364/OE.22.005147

1. Introduction

Since the optical phase can be retrieved from several interferograms easily, phase-shifting interferometry (PSI) has been widely used in wave-front reconstruction, deformation analysis, optics surface testing, etc [1]. To date, the reported phase retrieval algorithms are mainly consisted of the spatial domain algorithm and the temporal domain algorithm. In the spatial domain, Fourier transform based phase retrieval is a typical demodulation algorithm [2, 3], in which only if the fringe pattern exhibits the characteristics of narrow bandwidth, high carrier frequency and low noise, the measured phase can be retrieved from one interferogram conveniently. However, in the case of closed-fringe pattern, this algorithm does not work well. In the temporal domain, N-step phase-shifting algorithm has been widely employed (N≥3), in which the phase can be retrieved from several phase-shifting interferograms. In general, assuming the phase shifts is a fixed value, three phase-shifting interferograms are enough for phase retrieval, but if the phase shifts are changed, at least four phase-shifting interferograms are required to perform the phase retrieval. From the view of spectrum analysis, each phase-shifting algorithm is equivalent to a quadrature filter performing filtering on several phase-shifting interferograms, and then the background and one of two exponential terms of interferograms can be eliminated effectively [4, 5].

Usually, in the temporal domain, the accuracy of phase retrieval is affected by the environmental interference; moreover, the quality of the captured interferogram is greatly correlated with the performance of phase shifter. That is to say, in the practical PSI, it is still required to perform the phase retrieval from a few phase-shifting interferograms with unknown phase shifts. Aiming at the situation that the phase shifts are unknown, some phase retrieval algorithms have been proposed in the recent decade [6–9]. First, based on the least-square error estimation and the spatial alternative iteration, an advanced iterative algorithm (AIA) with high precision is proposed to perform the phase retrieval while it is greatly time-consuming [6]. In [7], from three interferograms with unknown phase shifts, a rapid phase retrieval algorithm is introduced while it is required that the unknown phase shifts should be fixed. In [8], from three phase-shifting shadow moiré patterns with unknown phase shifts, an accurate phase retrieval algorithm is presented, in which the spiral phase transform (SPT) is employed to determine the translation amount of grating and estimate the phase shifts, and then the phase retrieval is performed with the least-square iterative algorithm. Nevertheless, in both [7] and [8], it is required that the background should be eliminated in advance. In [9], a blind self-calibrating algorithm is proposed while it is also greatly time-consuming, in which the cross-bispectrum function is utilized to determine the relationship between the amplitude of reference wave and the measured phase. In addition, using the subtraction operation between interferograms to eliminate the background, the phase shifts can be determined with the standard deviation of difference maps by arc cosine function [10]. However, using this algorithm, in the case of the background noise exists, the result obtained with arc cosine function is the complex while the phase shift is close to π.

Recently, from two interferograms with unknown phase shift, some two-step phase retrieval algorithms have been reported [11–14]. In [11], a regularized optical flow algorithm and SPT are respectively utilized to determine the fringe direction of interference pattern and two phase-correlated quadrature signals, and then the measured phase is retrieved by arc tangent function. The advantage of this algorithm is robust against the additive noise and the variation of phase shifts. In [12], by searching for the minimum of merit function, a self-tuning (ST) algorithm is proposed to perform the phase shifts extraction. However, in this algorithm, while the phase shift is far from π/2, the accuracy of phase shifts extraction is also greatly decreased. In [13], using the maximum and the minimum of interference (EVI), an accurate two-step phase demodulation algorithm is presented. In [14], based on the Gram–Schmidt orthonormalization, another accurate two-step phase retrieval approach, which is named as GS algorithm, is also introduced, in which an orthonormal interferogram is determined by Gram–Schmidt orthonormalization of two interferograms. However, in all above two-step algorithms, it is still required to eliminate the background in advance. That is to say, the accuracy of phase retrieval is greatly correlated with the background, the smaller of background, and the higher accuracy of phase retrieval. To eliminate the background, it was reported that the object wave and the reference wave are respectively recorded by a shutter, and then the background noise is eliminated by the subtraction operation [15]. However, this approach will increase the complexity of experimental system and the recording time. In general, for the interferograms with slow background variation, both the spatial filtering algorithm [2, 16] and the spatial averaging technique [17, 18] are suitable for the background elimination, However, for the interferograms with rapid background variation, the above algorithms do not work well.

In this study, from three interferograms with unknown phase shifts, by combing two normalized difference maps and two-step phase retrieval algorithm, we intend to search for an accurate and rapid phase retrieval approach.

2. Principle

In PSI, the intensity distribution of a sequence of phase-shifting interferograms can be described as:

I_{n, k} = a_{k} + b_{k} \cos [φ_{k} + θ_{n}]

where n(n = 1,2,…,N) and k(k = 1,2,…,K) respectively denote the sequence number of phase-shifting interferograms and the pixel position, K is total pixels number of interferogram. a_k, b_k and φ_k represent the background intensity, the modulation amplitude and the measured phase, respectively. θ_n denotes the phase shifts of the nth interferogram. In this study, three phase-shifting interferograms are required, thus n = 1, 2, 3. For simplicity, we define θ₁ = 0, thus two difference maps between the 2th, 3th interferograms and the 1st interferogram can be respectively expressed as

S_{1, k} = I_{1, k} - I_{2, k} = 2 b_{k} \sin \frac{θ_{2}}{2} \cdot \sin (Φ_{k})

S_{2, k} = I_{1, k} - I_{3, k} = 2 b_{k} \sin \frac{θ_{3}}{2} \cdot \sin (Φ_{k} + Δ)

where

Φ_{k} = φ_{k} + \frac{θ_{2}}{2}, Δ = \frac{θ_{3} - θ_{2}}{2}

_.

From Eqs. (2) and (3), we can see that the background term a_k in Eq. (1) does not appear in the difference maps, indicating that the background has been eliminated by the simple subtraction operation. As we knows, in the practical PSI, the phase shifts between interferograms is changed, thus $2 b_{k} \sin \frac{θ_{2}}{2} \neq 2 b_{k} \sin \frac{θ_{3}}{2}$ , that is to say, the amplitude of difference map S_1,_k is different from S_2,_k. To eliminate the amplitude inequality of two difference maps, in this study, the normalizing approach is introduced to perform the normalization of two difference maps.

In general, the normalization of vector $u$ can be described as

\tilde{u} = u / \sqrt{〈 u, u 〉} = u / ‖ u ‖

Where $\tilde{u}$ represent the normalized vector, <•,•> represent the inner product operator.

Similarly, the normalization of S₁_,k (or S₂_,k) can be expressed as

〈 S_{i, k}, S_{i, k} 〉 = \sum_{k = 1}^{K} S_{i, k} \cdot S_{i, k}

Normalizing Eqs. (2) and (3), we have

{\tilde{S}}_{1, k} = b_{k} \cdot \sin (Φ_{k}) / \sqrt{\sum_{k = 1}^{K} b_{k}^{2} \cdot \sin^{2} (Φ_{k})}

{\tilde{S}}_{2, k} = b_{k} \cdot \sin (Φ_{k} + Δ) / \sqrt{\sum_{k = 1}^{K} b_{k}^{2} \cdot \sin^{2} (Φ_{k} + Δ)}

If the fringe number in the interferogram is more than one, the following approximation should exist

\begin{matrix} \sum_{k = 1}^{K} b_{k}^{2} \cdot \sin^{2} (Φ_{k}) - \sum_{k = 1}^{K} b_{k}^{2} \cdot \sin^{2} (Φ_{k} + Δ) \\ = \sum_{k = 1}^{K} b_{k}^{2} \cdot [\sin (Φ_{k}) - \sin (Φ_{k} + Δ)] \cdot [\sin (Φ_{k}) + \sin (Φ_{k} + Δ)] \\ = 4 \cdot \sum_{k = 1}^{K} b_{k}^{2} \cdot \sin (\frac{2 Φ_{k} + Δ}{2}) \cdot \cos (\frac{2 Φ_{k} + Δ}{2}) \cdot \sin (\frac{- Δ}{2}) \cdot \cos (\frac{- Δ}{2}) \\ = \sum_{k = 1}^{K} b_{k}^{2} \cdot \sin (2 Φ_{k} + Δ) \sin (- Δ) \\ \approx 0 \end{matrix}

And then we will have the following approximation

\sqrt{\sum_{k = 1}^{K} b_{k}^{2} \cdot \sin^{2} (Φ_{k})} \approx \sqrt{\sum_{k = 1}^{K} b_{k}^{2} \cdot \sin^{2} (Φ_{k} + Δ)}

Thus the above two normalized difference maps can be expressed as

{\hat{S}}_{1, k} = b_{k}^{'} \cdot \sin (Φ_{k}) = - b_{k}^{'} \cdot \cos (Φ_{k} + π / 2) = b_{k}^{''} \cdot \cos (Φ_{k}^{'})

{\hat{S}}_{2, k} = b_{k}^{'} \cdot \sin (Φ_{k} + Δ) = - b_{k}^{'} \cdot \cos (Φ_{k} + π / 2 + Δ) = b_{k}^{''} \cdot \cos (Φ_{k}^{'} + Δ)

where

b_{k}^{''} = - b_{k}^{'}

,

Φ_{k}^{'} = Φ_{k} + π / 2

.

As described in [2, 11–14], it is required to eliminate the background in advance for all two-step phase retrieval algorithms. Specially, in the case that the phase shifts is close to π, almost all reported two-step phase retrieval algorithms do not work well. From Eqs. (10) and (11), we can see that the background and the amplitude inequality have been eliminated in the normalized difference maps. Obviously, combining the two normalized difference maps and the two-step phase retrieval algorithm without background filtering, the accurate phase will be retrieved rapidly. Importantly, using the normalized difference maps, so long as the phase shifts do not reach 2π, it is impossible to appear the situation of $Δ = (θ_{3} - θ_{2}) / 2 = π$ , thus the phase with high precision also should be achieved.

3. Simulation

To verify the effectiveness of the proposed algorithm, firstly, we perform the phase retrieval by the simulation calculation. According to Eq. (1), three simulated fringe patterns with the size of 512?512 pixels are generated, in which the background intensity and the modulation amplitude are respectively set as $a (x, y) = 3 p e a k s (512) + 10 \exp [- 0.05 (x^{2} + y^{2})] + 80$ and $b (x, y) = 100 \exp [- 0.2 (x^{2} + y^{2})]$ . peaks denotes the peaks function in Matlab, which is a function of two variables and obtained by translating and scaling Gaussian distribution, is employed to generate the background with rapid variation. The measured phase is set as $φ (x, y) = π (1.5 x^{2} + 0.5 y^{3})$ , where $x, y \in [- 2.56, 2.56]$ , and the phase shifts are set as $θ_{1} = 0, θ_{2} = 1, θ_{3} = 2$ rad. In addition, Gaussian additive noise with the signal-to-noise ratio (SNR) of 3% and the random noise of 2% are added to the fringe pattern. In this study, we firstly choose two-step GS algorithm to perform the phase retrieval of two normalized difference maps. For convenience, we define the combination of the normalized difference maps and two-step GS algorithm without background filtering as GS3 approach, the conventional two-step GS algorithm with Gaussian high-pass background filtering as GS2 approach, and the two-step GS algorithm with the spatial averaging technique filtering as GSm approach. Figures 1(a) and 1(b) show one of three simulated fringe patterns and its theoretical phase, respectively. Using GS3, GS2 and GSm approaches, as shown in Figs. 1(c)–1(e), it is presented that the root mean square errors (RMSE) between the theoretical phase and the measured phase are respectively 0.03 rad, 0.15rad and 0.32rad, indicating the accuracy of phase retrieval with GS3 approach is greatly higher than that with GS2 approach, and the result with GSm approach is very bad. In our calculation, the transfer function of Gaussian high-pass filter can be expressed as $H (u, v) = 1 - \exp [- (u^{2} + v^{2}) / 2 σ^{2}]$ , in which $σ$ denotes the width of the filter window and is set as 1.5. Clearly, in the case of the rapid background variation, the spatial average technique is not suitable for the background elimination of interferogram. Therefore, in the following experimental research, we only perform the comparison of the phase retrieval between the proposed approach and the high-pass filtering approach.

Fig. 1 Simulation results. (a) one of the three simulated interferograms; (b) the theoretical phase; (c) the phase difference between the theoretical phase and the unwrapped phase obtained by GS3 approach; (d) phase difference between the theoretical phase and the unwrapped phase obtained with GS2 approach; (e) phase difference between the theoretical phase and the unwrapped phase obtained with GSm approach.

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As described in [11], in the case that the phase shift is larger than 2.5 rad, almost all two-step phase retrieval algorithms do not work well. To solve this problem, in this study, three fringe patterns are generated again, in which the phase shifts of the 1st frame and the 2th frame are respectively set as 0 and 0.8 rad while the phase shifts of the 3th frame is changed from 1.7 to 3.14 rad with the interval of 0.02 rad. As shown in Fig. 2, the relationship between the RMSE of phase retrieval with GS3, GS2 approaches and the phase shifts of the 3th frame is presented. Obviously, it can be seen that while the phase shift is lower than 2.5 rad, the RMSE of phase retrieval with GS3 approach is significantly lower than that with GS2 approach; moreover, even if the phase shifts is larger than 2.5 rad, the RMSE of phase retrieval with GS3 approach is still lower than 0.03rad while GS2 approach does not work well. That is to say, using the normalized difference maps, the condition requirement of two-step GS phase retrieval algorithm is decreased significantly.

Fig. 2 Relationship between the RMSE of phase retrieval with GS3, GS2 approaches and the phase shifts, respectively.

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4. Experimental research

Following, three frame experimental phase-shifting interferograms are utilized to verify the feasibility of the proposed approach, as shown in Fig. 3(a), in which the phase shifts of the 2th frame and the 3th frame are about 1 rad and 2 rad, respectively. And to eliminate the background effectively, the width of the filter window is set as 3. Using the proposed GS3 approach, Fig. 3(b) shows the wrapped phase. Like the simulation calculation, the wrapped phase obtained with GS2 approach is also presented (Fig. 3(c)), and the wrapped phase obtained with AIA of 60 phase-shifting interferograms is employed as the reference (Fig. 3(d)). Obviously, it is observed that the RMSE between the reference phase and the phase obtained with GS3, GS2 approaches are respectively 0.05 rad and 0.17rad, further indicating that the accuracy of phase retrieval with GS3 approach is indeed higher than that with GS2 approach.

Fig. 3 Experimental results. (a) one of three experimental phase-shifting interferograms; (b) the wrapped phase obtained with GS3 approach; (c) the wrapped phase obtained with GS2 approach; (d) the reference wrapped phase obtained by AIA.

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Subsequently, to verify the feasibility of the proposed approach, the other two-step algorithms are also utilized to perform phase retrieval of the normalized difference maps. Like the above definition of GS3, GS2 approaches, we also define ST3, ST2 approaches, EVI3, EVI2 approaches and Kreis3, Kreis2 approaches, in which the number 3 approach represent that the combination of the normalized difference maps and the corresponding two-step algorithm without background filtering while the number 2 approach represent the conventional two-step algorithm with Gaussian high-pass background filtering. Moreover, the phase obtained with AIA is also employed as the reference. As shown in Table 1, for each phase retrieval approach, its RMSE and Processing time are presented, respectively.

Table 1. RMSE, Processing Time with Different Approaches

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From Table 1, it is observed that RMSE, Processing time in all number 3 approaches are greatly lower than those in all number 2 approaches, indicating that the proposed normalized difference maps approach is suitable for all two-step phase retrieval algorithms. Specially, it is seen that the processing time in GS3 and EVI3 approaches are greatly lower than those in GS2 and EVI2 approaches, indicating the speed of background elimination with the difference maps is faster than that with Gaussian high-pass filtering.

Like the simulation analysis, if the phase shift is larger than 2.5 rad, using the proposed normalized difference maps approach, we also perform the experimental research of phase retrieval. First, three frame experimental phase-shifting interferograms, in which the phase shifts of the 2th frame and the 3th frame are respectively set as about 3 rad and 5 rad, are chosen to obtain two normalized difference maps. Second, combining the normalized difference maps and the two-step GS algorithm without background filtering, the measured phase can be achieved, as shown in Fig. (a). In contrast, the measured phases obtained with GS2 approach and AIA are respectively shown in Figs. 4(b) and 4(c). Obviously, it can be seen that while the phase shifts is larger than 2.5 rad, GS2 approach cannot work well. However, in our experiment, the actual phase shifts between two difference maps is only 1 rad, thus the accuracy of phase retrieval with GS3 approach is almost the same with AIA, indicating the proposed approach is suitable for the phase retrieval of arbitrary phase shifts.

Fig. 4 Unwrapped phases obtained with different approaches while the phase-shift is larger than 2.5rad. (a) GS3 approach; (b) GS2 approach; (c) AIA.

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5. Conclusion

In this study, combing two normalized difference maps induced by three interferograms with unknown phase shifts and two-step phase retrieval algorithms, the measured phase with high precision can be retrieved rapidly and easily. First, using the simple subtraction operation between interferograms, two difference maps without background can be achieved easily. Second, using the normalization process to eliminate the amplitude inequality of difference maps, two normalized difference maps can be obtained. Third, combining two normalized difference maps and two-step phase retrieval algorithm, the measured phase with high precision can be obtained rapidly. Both simulation calculation and experimental research show that the accuracy and processing time of phase retrieval with the proposed normalized difference maps are greatly better than those with the conventional two-step phase retrieval algorithm. Importantly, in the case that the phase shifts is close to π, almost all reported two-step phase retrieval algorithms do not work; however, using the proposed normalized difference maps approach, the measured phase with high precision also can be retrieved rapidly, indicating the proposed normalized difference maps approach is suitable for the phase retrieval with arbitrary phase shifts. In a word, this proposed normalized difference maps approach will supply a powerful tool for the phase retrieval with arbitrary phase shifts.

Acknowledgments

This work is supported by National Nature Science Foundation of China grants (61177005, 61275015, and 61078064).

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Approaches	RMSE /rad	Processing time/s	Approaches	RMSE /rad	Processing time/s
GS3	0.0346	0.0333	GS2	0.1669	0.1031
ST3	0.1061	2.2674	ST2	0.1704	2.3591
EVI3	0.0510	0.0537	EVI2	0.1695	0.1187
Kreis3	0.1121	0.3758	Kreis2	0.2889	0.4556

Phase retrieval approach based on the normalized difference maps induced by three interferograms with unknown phase shifts

Abstract

1. Introduction

2. Principle

3. Simulation

4. Experimental research

5. Conclusion

Acknowledgments

References and links

Cited By

Figures (4)

Tables (1)

Equations (11)

Optics Express