Improving the phase reconstruction accuracy of simultaneous phase-shifted lateral shearing interferometry using a polarization redundant sub-region interpolation method

Siqi Wang; Siqi Wang; Bingcai Liu; Bingcai Liu; Hongjun Wang; Hongjun Wang; Yahui Zhu; Yahui Zhu; Kai Wang; Kai Wang; Kexin Ren; Kexin Ren; Yuwen Zhang; Yuwen Zhang; Ailing Tian; Ailing Tian

doi:10.1364/OE.463534

1. Introduction

Generally, simultaneous phase-shifted lateral shearing interferometry systems, prism elements, diffractive elements, or micro polarization elements are usually used to achieve simultaneous phase shifts [1]. However, when using prism elements to achieve simultaneous phase shift, it is not easy to achieve complete synchronization because numerous cameras are used to acquire interferometric images. The collected simultaneous phase-shifted interferograms have position mismatch errors, resulting in additional calibration alignment efforts [2]. When using diffractive elements to achieve simultaneous phase shift, the utilization of optical energy is significantly reduced, and the acquired simultaneous phase shift interferograms also have spatial position mismatch errors [3]. A typical device that uses micro polarization elements to split light is the division of focal plane (DoFP) polarization camera, which has the advantages of high synchronization, compact system structure, and no need for alignment. It can meet the high synchronization requirements of the measurement system while significantly reducing the use of devices in the optical path system, thus avoiding unnecessary error sources to a greater extent. The DoFP polarization camera integrates a micro-polarizer array with four polarization direction (${0^ \circ }$, ${45^ \circ }$, ${90^ \circ }$, ${135^ \circ }$) units into the super-pixels of the focal plane array sensor, as shown in Fig. 1.

Fig. 1. (a) The array of charge-coupled device (CCD) imaging elements is covered with four pixelated linear polarization filters oriented at 0°, 45°, 90°, and 135°. (b) Micro-polarization array. (c) Super-pixel.

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Based on the advantages of DoFP polarization cameras, they have been widely used as phase-shift devices in various simultaneous phase-shifted interferometry systems in recent years [4–6]. Still, their inherent defective instantaneous field of view (IFoV) error has limited their optical measurement applications [7]. This error occurs because adjacent pixels have different line polarization directions, thus affecting the instantaneous field of view. When acquiring a simultaneous phase-shifted interferogram with a single frame, adjacent pixels with varying polarization directions will block each other, resulting in a significant reduction in the spatial resolution [8]. A series of interpolation reconstruction methods have been proposed to solve the problem of missing solutions in the acquired DoFP images. These methods can be broadly divided into two categories: the first category is based on spatially invariant non-adaptive interpolation methods, which include traditional interpolation methods such as bilinear [8], bicubic [9], and bicubic spline [10]. This type of interpolation method was first proposed and applied to the interpolation reconstruction of DoFP polarization images, which can quickly estimate the light intensity response in the unknown polarization direction from the neighboring pixels. Unfortunately, the interpolation process does not consider the correlation and structural information between the polarization channels, resulting in significant errors in the reconstructed DoFP images. The other category is the adaptive interpolation method, which refers to improvements to the traditional method based on the correlation between the polarization channels. These include interpolation algorithms based on gradient information [11], polarization inter-channel correlation [12], and smoothness threshold [13]. All three algorithms introduce the determination of the image detail edge direction based on the traditional interpolation method. They propose and verify that interpolation along the edge direction can improve the reconstruction accuracy of detailed high-frequency regions. In 2017, Ahmed et al. [14] combined residual interpolation and guide filtering to estimate the unknown light intensity response. However, the guide image used in this method was an actual high-resolution image that could not be captured by the DoFP polarization camera, which limited its application. In 2018, Li [15] et al. first proposed polarization image edge determination and Newton polynomial-based interpolation in the polarization difference domain. In 2021, Wu [16] et al. expanded the polarization difference domain to an interpolation model with prior weighting parameters to improve the accuracy of the resolution reconstruction of DoFP images.

Based on the inspiration of the above various interpolation algorithms, combined with the characteristics of the simultaneous phase-shifted interferogram, the PRSI method is proposed to solve the reduced phase reconstruction accuracy of simultaneous phase-shifted lateral shearing interferometer due to the missing pixel points. The optimization of most current reconstruction methods is based on the consideration of inter-polarization channel correlation. However, a large number of related studies have demonstrated that not all the regions to be reconstructed have robust inter-channel correlations. Therefore, it is necessary to deal with the strong and weak inter-channel correlation parts in a sub-region.To achieve sub-regional processing, the method sets a threshold for the calculated polarization redundancy error value. Subsequently, a fast convolutional bilinear method is used to interpolate the smoothed regions, and a refined interpolation is taken for the stripe detail regions using the optimized edge weights. The PRSI method guarantees interpolation accuracy and low complexity while minimizing strong jagged artifacts in the stripe detail region due to intensity domain shifts. The phase reconstruction accuracy of the simultaneous phase-shifted lateral shearing interferometry is further improved.

The rest of the paper is organized as follows: In Section 2, the optical path and Jones matrix derivation of the simultaneous phase-shifted lateral shearing interferometry system are presented. Then, in Section 3, the proposed PRSI reconstruction method is described in detail. In Section 4, the proposed PRSI method is fully compared and validated with existing mainstream interpolation methods in simulated and experimental interferograms based on image metrics and optical metrics, respectively. Finally, the conclusions are drawn in Section 5, and the directions for the subsequent improvement and refinement of the technique are indicated.

2. Simultaneous phase-shifted lateral shearing interferometry

Measuring optical components by simultaneous phase-shifted lateral shearing interferometry has apparent advantages such as a common optical path, good interference fringes even in the absence of anti-oscillation, real-time dynamic measurement concerning the workpiece, and no need for a reference surface (self-comparing interference) [6]. Based on the particular advantages of this measurement method, a generalized fast optical component face shape detection method based on birefringent crystal theory, and simultaneous phase shift interferometry principle is proposed [17]. The optical path for the system is shown in Fig. 2.

Fig. 2. System optical path diagram of a generalized fast optical component face shape detection with simultaneous phase shift lateral shearing interferometry device.

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The basic principle of this optical system is that the laser beam of 632.992 nm emitted from the He-Ne laser passes through the beam spreading collimation system, and is turned by the reflector to be incident vertically on the BS, then reaches the surface of the original to be measured through the standard mirror. The wavefront to be calculated returned from the optical surface is incident vertically on the parallel polarization beam splitter through the deflector. Since the parallel polarization beam splitter has birefringence characteristics and can make the $o$-light and $e$-light produce a certain transverse displacement, the wavefront to be measured after the parallel polarization beam splitter becomes two beams of linearly polarized light with mutually orthogonal vibration direction. The parallel polarization beam splitter shear separates the two beams. After the fast axis direction and the two beams polarization direction angle of ${45^ \circ }$ QWP, the incident two beams of linear polarization into left-rotation circular polarization and right-rotation circular polarization light. The circularly polarized light arriving at the DoFP polarization camera can be expressed in terms of the Jones matrix as:

(1)$$\left\{ {\begin{array}{{c}} \begin{array}{c} {E_o} = {G_{{\lambda / 4}}} \cdot {P_o} \cdot M = \frac{1}{{\sqrt 2 }}\left[ {\begin{array}{{cc}} 1&i\\ i&1 \end{array}} \right] \cdot \left[ {\begin{array}{{c}} {\cos \theta }\\ 0 \end{array}} \right]\exp \left[ {iW(x,y)} \right]\\ = \frac{{\cos \theta \cdot \exp \left[ {iW(x,y)} \right]}}{{\sqrt 2 }}\left[ {\begin{array}{{c}} 1\\ i \end{array}} \right] \end{array}\\ \begin{array}{c} {E_e} = {G_{{\lambda / 4}}} \cdot {P_e} \cdot M = \frac{1}{{\sqrt 2 }}\left[ {\begin{array}{{cc}} 1&i\\ i&1 \end{array}} \right] \cdot \left[ {\begin{array}{{c}} 0\\ {\sin \theta } \end{array}} \right]\exp \left[ {iW(x - s,y)} \right]\\ = \frac{{\sin \theta \cdot \exp \left[ {iW(x - s,y)} \right]}}{{\sqrt 2 }}\left[ {\begin{array}{{c}} { - 1}\\ i \end{array}} \right] \end{array} \end{array}}. \right.$$

In the above equation, $M$ is the polarizer in the optical path, ${P_o}$ and ${P_e}$ are two beams of $o$-light and $e$-light in orthogonal directions passing through a parallel polarization beam splitter with the shear amount s, ${G_{{\lambda / 4}}}$ is a quarter-wave plate with a specific orientation, $W(x,y)$ is the wavefront information of the optical element to be measured, and ${E_o}$, ${E_e}$ are the two beams of circularly polarized light that finally reach the DoFP polarization camera. When the light enters the DoFP polarization camera, it passes through the micro-polarizer array ${N_\alpha }$ integrated inside the camera, which $\alpha$ is in four different polarization directions (${0^ \circ }$, ${45^ \circ }$, ${90^ \circ }$, ${135^ \circ }$). Specifically, the Jones matrix can be expressed as:

(2)$$\left\{ {\begin{aligned} \begin{aligned} {{E^{\prime}}_o} &= {N_\alpha } \cdot {E_o} = \left[ {\begin{aligned} {{{\cos }^2}\alpha }\quad{\sin \alpha \cos \alpha }\\ {\sin \alpha \cos \alpha }\quad{{{\sin }^2}\alpha } \end{aligned}} \right]\frac{{\cos \theta \cdot \exp [{iW(x,y)} ]}}{{\sqrt 2 }}\left[ {\begin{aligned} 1\\ i \end{aligned}} \right]\\& = \frac{{\cos \theta \cdot \exp [{i(W(x,y) + \alpha } ]}}{{\sqrt 2 }}\left[ {\begin{aligned} {\cos \alpha }\\ {\sin \alpha } \end{aligned}} \right] \end{aligned}\\ \begin{aligned} {{E^{\prime}}_e} &= {N_\alpha } \cdot {E_e} = \left[ {\begin{aligned} {{{\cos }^2}\alpha }\quad{\sin \alpha \cos \alpha }\\ {\sin \alpha \cos \alpha }\quad{{{\sin }^2}\alpha } \end{aligned}} \right]\frac{{\sin \theta \cdot \exp [{iW(x - s,y)} ]}}{{\sqrt 2 }}\left[ {\begin{aligned} { - 1}\\ i \end{aligned}} \right]\\& = \frac{{ - \sin \theta \cdot \exp [{iW(x - s,y) - \alpha } ]}}{{\sqrt 2 }}\left[ {\begin{aligned} {\cos \alpha }\\ {\sin \alpha } \end{aligned}} \right] \end{aligned} \end{aligned}}. \right.$$

Four simultaneous phase-shifted lateral shearing interferograms with a phase shift difference ${\pi / 2}$ are obtained using the phase shift array of the micro polarizer on the DoFP polarization camera. These four interferograms are precisely the low-resolution images to be subsequently interpolated and reconstructed, using the Jones matrix representation as follows:

(3)$$\left\{ \begin{array}{l} {I_1} = {|{{{E^{\prime}}_{ox,{0^ \circ }}} + {{E^{\prime}}_{ex,{0^ \circ }}}} |^2} = \frac{1}{2}(1 + \sin 2\theta \cos (\Delta {W_x}))\\ {I_2} = {|{{{E^{\prime}}_{ox,{{45}^ \circ }}} + {{E^{\prime}}_{ex,{{45}^ \circ }}}} |^2} = \frac{1}{2}(1 + \sin 2\theta \cos (\Delta {W_x} + \frac{\pi }{2}))\\ {I_3} = {|{{{E^{\prime}}_{ox,{{90}^ \circ }}} + {{E^{\prime}}_{ex,{{90}^ \circ }}}} |^2} = \frac{1}{2}(1 + \sin 2\theta \cos (\Delta {W_x} + \pi ))\\ {I_4} = {|{{{E^{\prime}}_{ox,{{135}^ \circ }}} + {{E^{\prime}}_{ex,{{135}^ \circ }}}} |^2} = \frac{1}{2}(1 + \sin 2\theta \cos (\Delta {W_x} + \frac{{3\pi }}{2})) \end{array} \right.,$$

(4)$$\Delta {W_x} = W(x,y) - W(x - s,y) = \arctan \left( {\frac{{{I_4} - {I_2}}}{{{I_3} - {I_1}}}} \right).$$

Finally, the four-step phase-shifting algorithm performs the phase reconstruction of the four simultaneous phase-shifted lateral shear interferograms after interpolation processing [18]. The $\Delta {W_x}$ is obtained as the differential phase of the wavefront information to be measured and the shear wavefront information, which can be further brought into the differential Zernike algorithm to solve the phase information of the optical element to be measured [19]. ${I_1}$, ${I_2}$, ${I_3}$, ${I_4}$ in Eq. (4) are high-resolution interferograms, at which time the inherent one-pixel point phase reconstruction error has been eliminated.

3. Polarization redundancy sub-region interpolation

3.1 Resolution reconstruction strategy

The DoFP polarization camera makes a trade-off between spatial resolution and temporal resolution. A high temporal resolution simultaneous phase shift interferogram is acquired while losing three-quarters of the spatial resolution. The missing stripe detail information of the synchronous phase shift interferogram is generally reconstructed by interpolation. The complete resolution reconstruction process is as follows: In the first step, the low-resolution subgraphs of four different polarization directions are obtained by down-sampling the original images acquired by the DoFP polarization camera. The second step performs the corresponding sparse channel decomposition for the four sub-maps according to their aligned positions in the micro-polarization array. In the third step, the sparse images with the missing resolution are reconstructed using the PRSI method to obtain four high-resolution simultaneous phase-shifted interferograms. Figure 3 visualizes the whole spatial resolution interpolation reconstruction process of simultaneous phase-shifted interferograms.

Fig. 3. Simultaneous phase shift interferogram spatial resolution interpolation reconstruction flow chart.

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3.2 Polarization redundancy sub-regional approach

The innovation of using an ideal polarization redundancy relationship for sub-regional processing was inspired by the Accuracy of Polarization Measurement Redundancy (APMR) metric proposed by Li [20] et al. APMR is different from peak signal-to-noise ratio (PSNR), structural similarity index (SSIM) and mean square error (RMSE). This metric can predict the quality of DoFP polarization images without a reference image simply by the deviation between itself and the ideal state polarization redundancy value.

Since the polarization information of the object cannot be truly captured, the intensity is generally recorded and converted into the polarization information we need by calculating the Stokes parameters. Three of the commonly used basic Stokes parameters are defined as:

(5)$$\left\{ {\begin{array}{{c}} {{S_0} = 0.5\ast ({I_0} + {I_{45}} + {I_{90}} + {I_{135}})}\\ {{S_1} = {I_0} - {I_{90}}}\\ {{S_2} = {I_{45}} - {I_{135}}} \end{array}} \right.,$$

where ${I_0}$, ${I_{45}}$, ${I_{90}}$, and ${I_{135}}$ represent the light intensity values in the four polarization directions (${0^ \circ }$, ${45^ \circ }$, ${90^ \circ }$, ${135^ \circ }$). ${S_0}$, ${S_1}$, and ${S_2}$ are known as the first three Stokes parameters. The ${S_0}$ is the total intensity of the light. Then ${S_1}$, ${S_2}$ describe the amount of linear polarization [21].

As mentioned above, to obtain the Stokes vector parameter values, the most widely used method is to invert the corresponding Stokes vector values by getting the intensity values corresponding to the four different polarization directions, where the following equation can calculate the intensity values:

(6) $$I = TS,$$

where $T$ is the measurement matrix $4 \times 3$ defined as follows:

(7)$$T = \frac{1}{2}\left[ {\begin{array}{{ccc}} 1&1&0\\ 1&0&1\\ 1&{ - 1}&0\\ 1&0&{ - 1} \end{array}} \right].$$

The ideal simultaneous phase-shifted interferograms should satisfy the following relationship based on Eq. (6) and Eq. (7).

(8)$${I_0} + {I_{90}} = {I_{45}} + {I_{135}} = {S_0}.$$

The above relationship shows that measuring the intensity values in just three different polarization directions is sufficient to estimate the total linear Stokes vector values. Therefore, the intensity value of the fourth polarization direction is redundant information, and this redundant information allows Eq. (8) to hold, for the ideal simultaneous phase-shifted interferograms should satisfy ${I_0} + {I_{90}} - {I_{45}} - {I_{135}} = 0$. However, the lack of resolution or interpolation accuracy in an actual scene makes the above equation challenging to hold.

So the pixel position $(i,j)$, at which i, $j$ satisfies $i \in [1,h],j \in [1,w]$, where w and h denote the width and height of the image. The measured intensity values can be written as:

(9)$${\hat{I}_\theta }(i,j) = {I_\theta }(i,j) + {\varepsilon _\theta }(i,j),\theta = {0^\circ },{45^\circ },{90^\circ },{135^\circ },$$

where ${\hat{I}_\theta }(i,j)$ is the ideal intensity measurement result, and ${\varepsilon _\theta }(i,j)$ is the error value due to the lack of resolution. Based on Eq. (8) and Eq. (9), we can obtain:

(10)$$\begin{aligned} {R_{i,j}} &= {{\hat{I}}_0}(i,j) + {{\hat{I}}_{90}}(i,j) - {{\hat{I}}_{45}}(i,j) - {{\hat{I}}_{135}}(i,j)\\& = {\varepsilon _0}(i,j) + {\varepsilon _{90}}(i,j) - {\varepsilon _{45}}(i,j) - {\varepsilon _{135}}(i,j). \end{aligned}$$

The ${R_{i,j}}$ is called the redundancy error value, and the value should be close to zero for high-quality simultaneous phase-shifted interferograms. Still, the redundancy error is always non-zero due to the lack of resolution. There is often an enormous redundancy value in the high-frequency stripe detail region in simultaneous phase-shifted interferograms. Figure 4 visualizes the redundancy error values at low spatial resolution and high spatial resolution.

Fig. 4. Visualization of redundancy error values. (a) Low-resolution simultaneous phase-shifted interferograms absolute redundant error $|{R_{i,j}}|$. (b) High-resolution simultaneous phase-shifted interferograms with absolute redundant error $|{R_{i,j}}|$.

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In general sub-regional algorithms, the speed and direction of gradient changes are usually used to distinguish regions. To further improve the performance of gradient [11] and smoothness threshold [13] based on sub-region, the method of sub-region using ideal polarization redundancy relation is proposed. This method can solve the problem of poor sub-region effect due to the uncertainty of edge direction in the sub-region process. It also avoids the repetitive operation problem of applying multiple sub-regional algorithms for interferograms with different polarization directions. The basic process of this sub-region method is: In the first step, a suitable redundancy sub-region threshold is selected based on the empirical and experimental validation results. The threshold value is brought into the redundancy error formula for calculation in the second step. In the third step, if the calculated redundancy value is less than the selected threshold, the region is judged to be a smooth region, and on the contrary, the part is deemed a streak detail region. The specific process is as follows:

1. Preliminary predictive interpolation of four simultaneous phase-shifted interferograms using a fast convolution bilinear method.
2. Bring the interpolated simultaneous phase-shifted interferograms into Eq. (10) to obtain the redundant error value.
3. Set the redundant sub-region threshold ${Q_R}(i,j)$. If the redundant error exceeds the threshold, it will be judged as a high-redundancy error area ${P_H}$, namely the edge stripe area. Otherwise, it will be regarded as a low-redundancy error area ${P_L}$, namely a background or smooth area, which is specifically defined as follows: $(11)$${Q_R}(i,j) = \frac{1}{\delta }\min (\frac{{|{{R_{i,j}}} |- \min (|R |)}}{{\max (|R |) - \min (|R |)}},\delta ),$$$ $(12)$$\left\{ {\begin{array}{{c}} {{R_{i,j}} \ge {Q_R}(i,j),{P_H}}\\ {{R_{i,j}}\mathrm{\ < }{Q_R}(i,j),{P_L}} \end{array}}. \right.$$$

In Fig. 5(a), we visualize the subregion effect of the simultaneous phase-shifted interferogram with different hyperparameter settings in the X and Y shear directions using pseudo-colors. It can be visualized from the figure that the subregion effect is better when the parameter value is around 0.01. To further determine the optimal parameter selection interval, we randomly selected 20 low-resolution simultaneous phase-shifted interferograms corresponding to different face shapes in the established data set. The average results of the 20 interferograms (see Fig. 5(b)) show that the PSNR values are higher when the parameter values are between 0.01 and 0.015. The PSNR value is chosen here as the reference value for evaluating the reconstruction accuracy because the image metrics are easier to obtain, and in the subsequent study, we also found that the accuracy of the final phase reconstruction is positively proportional to the PSNR value of the interferogram after sub-regional interpolation.

Fig. 5. Effect of parameter $\delta$ adjustment on sub-regional results and reconstruction effect. (a) Visualization of the effect of parameter $\delta$ adjustment on sub-region. (b) Effect of parameter $\delta$ adjustment on reconstruction effect.

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Based on the above discussion, we set the parameter value $\delta$ to 0.01 by default in all subsequent experimental comparisons and validations. Of course, suppose the sub-regional model will be applied to other kinds of polarized images. In that case, the optimal value $\delta$ can be set according to the effect of the sub-region and verification of experimental results to achieve a better sub-region effect.

3.3 Optimizing edge weights

Based on the observation of polarization differential channel characteristics, we found a significant correlation between orthogonal and non-orthogonal polarization differential channels when crossing the streak detail region. The relationship between the different differential channels was theoretically analyzed and proved in the literature [16]. The paper first selected four typical edge cases from the high-resolution polarization image dataset acquired by the DoT detection system [22]. Secondly, the difference between the ${0^ \circ }$ polarization channel and the other three channels was used as an example to record the variation of pixel values when the three different images crossed the edge region from different directions. Finally, the authors demonstrate a specific correlation of the differential values between the orthogonal polarization channels and the non-orthogonal ones at high frequencies by deriving the equations.

In the literature [16], the authors define the global weight ratio of non-orthogonal and orthogonal difference channels as $1:\sqrt 2$. However, this ratio proposed in the paper is only applicable to the global image and not to the smooth or detailed regions of the image alone. The article also pointed out that this ratio is mainly $1:1$ in the background region of DoFP polarized images when validated on several datasets. Still, the inter-channel difference ratio in the edge region is not mentioned.

In this paper, the ideal synchronous phase shift interferogram is tested and verified, as shown in Fig. 6, which shows the variation of the difference between the ${0^ \circ }$ polarization channel and the other three channels when crossing the fringe region from different angles. The apparent correlation between the different values of the orthogonal and non-orthogonal polarization channels when traveling the fringe region in different directions is also verified. Of course, the figure shows only the differential correlation between the ${0^ \circ }$ polarization channel and the other three polarization channels due to space limitations. Similar relationships between the different correlations of the other polarization channels are found in the subsequent verification.

Fig. 6. Variation of the differential value when passing through the detail fringe region of the simultaneous phase-shifted interferogram at different angles.

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We define the difference values between different channels in the streak edge region as ${\Delta ^{{\theta _i},{\theta _j}}}$ satisfying $i \ne j$, and ${\theta _1},{\theta _2},{\theta _3},{\theta _4} \in ({0^ \circ },{45^ \circ },{90^ \circ },{135^ \circ })$. The value of the weight corresponding to the difference of the orthogonal polarization channel when passing through the edge of the streak in different directions is specified as ${w_\theta }$, the weight value corresponding to the non-orthogonal polarization channel difference is ${{(1 - {w_\theta })} / 2}$, which is expressed as follows:

(13)$${w_{{\theta _1},{\theta _2}}} = \left\{ {\begin{array}{{cl}} {w_\theta } = \frac{{{{\Delta }^{{\theta _1},{\theta _2}}}}}{{{{\Delta }^{{\theta _1},{\theta _2}}} + {{\Delta }^{{\theta _1},{\theta _3}}} + {{\Delta }^{{\theta _1},{\theta _4}}}}},&{\theta _1},{\theta _2},{\theta _3},{\theta _4} \in \theta ,\left| {{\theta _1} - {\theta _2}} \right| = {{90}^\circ }\\ \textrm{}(1 - {w_\theta })/2,&{\theta _1},{\theta _2},{\theta _3},{\theta _4} \in \theta ,\left| {{\theta _1} - {\theta _2}} \right| \ne {{90}^\circ } \end{array}}. \right.$$

According to the formula mentioned above for the different weights between orthogonal polarization channels, the weights between orthogonal channels under other polarization channels along different directions through the fringe region of the ideal synchronous phase shift interferogram are brought into the solution respectively. The optimal edge weights between the orthogonal polarization difference channels obtained above are recorded in Table 1.

Table 1. The optimal edge weight value of orthogonal channel difference

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According to the experimental statistics, the calculated orthogonal direction channel difference weight ${w_\theta }$ is stable at around 0.4982, while the non-orthogonal direction channel difference weight ${{(1 - {w_\theta })} / 2}$ is also stable at 0.2509. The above actual simultaneous phase shift interferogram statistics are used as the optimal fringe weight values for the subsequent interpolation reconstruction of the fringe regions.

3.4 Complete interpolation reconstruction process

The details of the interpolation reconstruction using the PRSI method can be described in the following steps:

1. The original image acquired by the DoFP polarization camera is defined as ${I_{DoFP}}$, and the four simultaneous phase-shifted interferograms extracted from the original image are defined as ${\hat{I}_\theta }(\theta = {0^ \circ },{45^ \circ },{90^ \circ },{135^ \circ })$. $(14)$${\hat{I}_\theta } = {I_{DoFP}} \odot mask_{i,j}^\theta ,$$$

where ${\odot}$ is the Hadamard operator symbol, and $mask_{i,j}^\theta$ represents the binary matrix at the pixel position $(i,j)$, which is defined as follows:

(15)$$mask_{i,j}^\theta = \left\{ {\textrm{ }\begin{array}{{c}} {1\textrm{ },i,j \in \theta }\\ {0\textrm{ },i,j \notin \theta } \end{array}}. \right.$$

1. The initial estimation of each channel is performed by bilinear interpolation. Here we use convolution to implement bilinear interpolation as follows quickly: $(16)$${\tilde{I}_\theta } = {\hat{I}_\theta } \ast F,$$$ where ${\ast} $ is the convolution operator and the bilinear interpolation filter F being defined as: $(17)$$F = \frac{1}{4}\left[ {\begin{array}{{ccc}} 1&2&1\\ 2&4&2\\ 1&2&1 \end{array}} \right].$$$
2. Where ${\theta _1},{\theta _2}$ represent the two differential channels calculated, and the channel differential values ${\hat{\varDelta }^{{\theta _1},{\theta _2}}}$ are calculated as follows: $(18)$${\hat{\varDelta }^{{\theta _1},{\theta _2}}} = ({\hat{I}_{{\theta _1}}} - {\tilde{I}_{{\theta _2}}} \odot mas{k_{{\theta _1}}}) \ast F.$$$
3. Based on the optimized edge weights ${w_{{\theta _1},{\theta _2}}}$ obtained in the previous section, the interpolation process for the edge part of ${\theta _1}$ can be expressed as: $(19)$${I_{{\theta _1}}} = \sum\limits_{{\theta _1} \ne {\theta _2}} {{w_{{\theta _1},{\theta _2}}}} (\tilde{I} + {\hat{\varDelta }^{{\theta _1},{\theta _2}}}).$$$
4. If the region is determined to be an edge region (${P_H}$) by the sub-region method in Section 3.2, then the area is reconstructed by using the optimal edge weights for differential domain fusion weighted interpolation; if the result is determined to be a smooth region (${P_L}$), then the part is reconstructed by using fast convolution bilinear interpolation. Finally, the two results are summed to obtain a high-resolution simultaneous phase-shifted interferogram on a single polarization channel ${I_{{\theta _1}}}$, which is defined as follows. $(20)$${I_{{\theta _1}}} = {I_{{\theta _1}}}({P_L}) \ast F + {I_{{\theta _1}}}({P_H})\sum\limits_{{\theta _1} \ne {\theta _2}} {{w_{{\theta _1},{\theta _2}}}} (\tilde{I} + {\hat{\varDelta }^{{\theta _1},{\theta _2}}}).$$$

Instead of using the proposed global differential a priori weights, a new optimal fringe weight is presented for the simultaneous phase-shifted interferogram. The idea is based on the consideration that the channel difference variation in the stripe detail region can be summarized and inferred from the simulation and test data of multiple high-resolution simultaneous phase-shift interferograms in an ideal situation. The proposed optimal fringe weights have good applicability in reconstructing high-resolution simultaneous phase-shifted interferograms. Figure 7 shows the interpolation reconstruction process of the ${0^ \circ }$ channel simultaneous phase-shifted interferograms, and the reconstruction process of other channels is similar.

Fig. 7. ${I_0}$ polarization channel spatial resolution reconstruction process using the PRSI method.

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4. Results and discussion

4.1 Resolution reconstruction results

This paper proposes the PRSI method for the acquired simultaneous phase shift interferogram. The peak signal-to-noise ratio (PSNR) and root mean square error (RMSE) are used to evaluate and verify the interferogram quality after interpolation reconstruction. According to the evaluation index in the field of image super-resolution reconstruction, in general, if the PSNR value is higher or the RMSE value is lower, it means that the interpolated reconstructed image is closer to the real high-resolution simultaneous phase shift interferogram, which means that the interpolated reconstructed image by this method has better visual reconstruction effect.

The PRSI method proposed in this paper is compared with the traditional interpolation methods, including bilinear interpolation (BL) [8], bicubic cubic interpolation (BC) [9], bicubic spline interpolation (BS) [10], and several more advanced interpolation methods proposed in recent years including Newton polynomial interpolation (NP) [15], polarization channel difference prior (PCDP) [16] and edge-aware residual interpolation (EARI) [22] for comparison. To make the data more convincing during the image metric calculation, we take the average of the results of multiple simultaneous phase shift interferograms calculated under different interpolation methods as the final measured results of this method for recording.

We visualize the absolute redundancy errors of the X- and Y-direction simultaneous phase-shifted lateral shearing interferograms after applying different interpolation algorithms using pseudo-colors, and the absolute redundancy errors of all interpolated interferograms are expanded to ten times the original values for easy observation. In Fig. 8, it can be observed that the proposed PRSI method has smaller absolute redundancy error values than other interpolation methods, especially in the processing of the fringe part, and the visual reconstruction effect of this interpolation method is significantly better than other interpolation methods. Table 2 compares the PSNR and RMSE values of different algorithms with the proposed algorithm. The method retains the low complexity of the traditional interpolation algorithm and suggests the optimal edge weights applicable to reconstruct the spatial resolution of the interferogram stripe detail region. Based on the above two innovations enables the interpolation method to achieve a better balance between algorithm running time and interpolation accuracy.

Fig. 8. Comparison of absolute redundancy error |R| visualization of different interpolation algorithms. (a) Real high-resolution image. (b) BL method. (c) BC method. (d) BS method. (e) PCDP method. (f) EARI method. (g) NP method. (h) PRSI method.

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Table 2. Comparison of PSNR and RMSE test results

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4.2 Simulated phase reconstruction results

The PRSI method proposed in this paper aims to solve the problem of the inherent lack of resolution of interferograms acquired by the DoFP polarization camera. Based on the application context of detecting the face shape of optical elements using simultaneous phase-shifted lateral shearing interferometry, this paper focuses on comparing the phase reconstruction accuracy after processing using various interpolation reconstruction methods. The specific validation process includes: in the first step, eight simultaneous phase-shifted interferograms in the X-shear and Y-shear directions are inverted by setting parameters such as curvature and eccentricity, and brought into the phase reconstruction algorithm as the original face shape to solve for the corresponding PV and RMS values [18,19]. In the second step, the four high-resolution interferograms are downsampled using the alignment position of the micro polarization array within the DoFP polarization camera as a guide. In the third step, different interpolation reconstruction methods are applied to reconstruct the downsampled four interferograms. In the fourth step, the interferograms after interpolation reconstruction are brought into the four-step phase-shift and differential Zernike algorithm, solved to obtain the corresponding PV and RMS values, and the obtained fitting results are compared with the original surface shapes.

In the simulation experiments, the peak function is introduced to add surface errors and ripples in the surface shapes with different curvatures c and eccentricities e to construct the medium frequency information. The high-frequency information is constructed by adding Gaussian noise with a variance of 0.01. Then, the corresponding simultaneous phase-shifted interferogram is inferred from the simulated surface shape information. The resulting interferograms are downsampled and subsequently reconstructed using different interpolation methods to obtain high-resolution interferograms with different shear directions. Bringing it into the four-step phase-shifted and differential Zernike algorithm for phase reconstruction, the obtained fitted and residual surface shape results are shown in Fig. 9.

Fig. 9. Surface shape fitting results after PRSI interpolation. (a1)-(c1) Simultaneous phase-shifted interferogram in the X-shear direction. (a2) Low-frequency fitted surface shape. (a3) Face shape residuals of (a2). (b2) Medium-frequency fitted surface shape. (b3) Face shape residuals of (b2). (c2) High-frequency fitted surface shape. (c3) Face shape residuals of (c2).

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The results of the reconstructed surface shapes using different methods are fully recorded in Table 3. Since the direct use of the PV and RMS values of the fitted surface shapes after interpolation to verify the interpolation methods does not provide a visual representation of the differences between the individual interpolation methods, the accuracy of the phase reconstruction is expressed as the difference between the above two in the following table. The wavelength unit used in the following equation is 1 λ=632.992 nm, which is defined as follows:

(21)$$\left\{ {\begin{array}{{c}} {\varDelta PV = |{P{V_{original}} - P{V_{fitted}}} |}\\ {\varDelta RMS = |{RM{S_{original}} - RM{S_{fitted}}} |} \end{array}}. \right.$$

Table 3. Comparison of $\varDelta PV$ and $\varDelta RMS$ value test results

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Comparing the data in the table, it can be found that the proposed PRSI method achieves the best performance in both low and medium-frequency face shapes. However, the proposed method does not perform as well as the EARI algorithm in the noise-added high-frequency face shape. The reason is that too much noise in the process of sub-region and reconstructing high redundancy regions can cause stronger disturbance to the PRSI reconstruction method. In the subsequent algorithm improvement process, adaptive parameter adjustment under noise or error scenarios can be added to the PRSI method, which makes the algorithm have better applicability. Overall, PRSI more efficiently solves the problem of significant degradation in phase reconstruction accuracy resulting from the use of DoFP polarization cameras in simultaneous phase-shifted lateral shearing interferometers.

4.3 Experimental phase reconstruction results

We built the simultaneous phase shift lateral shearing interferometer proposed in Section 2 on the experimental optical platform. The devices used to build the measurement system include a stable HRS015B model He-Ne laser (λ = 632.992 nm) from Thorlabs, a 40-magnification microscope objective (Microscope Objective), a 5 µm pinhole (GCO-P05A), a beam splitter (BS) (DCL-022P), a linear polarizer (LPVIS100) with an extinction ratio of 800:1, a shear device (Parallel Polarizing Beam Splitter) with a beam splitting distance of 1 mm made of Calcite material, a zero-stage quarter-wave plate (QWP) with the model name AQWP10-VIS, and a DoFP polarization camera with the model name Mako G-508B POL equipped with a Sony IMX250MZR chip.

First, the concave reflector (50.8 mm aperture, 400 mm focal length) was measured using a ZYGO interferometer, and the measurement results were saved as the true value of the face shape data (see Fig. 10(d)). Next, eight low-resolution simultaneous phase-shift interferograms of the experimentally acquired X and Y shear directions were reconstructed using several methods mentioned above, respectively. In Fig. 10(a), four reconstructed high-resolution interferograms corresponding to the X-shear direction are shown. The reconstructed interferograms are further brought into the phase reconstruction algorithm to obtain the surface shape fitting results, where the surface shape fitting results after PRSI reconstruction are shown in Fig. 10(c). The third column in Table 4 records the results of the surface shape fitted using the uninterpolated reconstructed low-resolution interferogram. A 5 µm pinhole was added to the experimental optical path, while median filtering denoising was performed on the acquired interferograms, to reduce the degree of noise contained in the interferograms as much as possible. In the preliminary experimental verification, it is found that the PRSI method is still the best choice to improve the phase reconstruction accuracy of the simultaneous phase-shifted lateral shearing interferometer in low-noise scenarios.

Fig. 10. Experimental acquisition and surface shape fitting results. (a) The experimental optical path built according to Fig. 2. (b) Simultaneous phase-shifted interferogram in X-shear direction after PRSI processing. (c) Fitting results of surface shape after PRSI reconstruction. (d) ZYGO interferometer test results.

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Table 4. Comparison of shape-fitted results after PRSI reconstruction

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5. Conclusion

This paper focuses on solving the problem of resolution deficit inherent to the DoFP polarization camera in simultaneous phase-shifted lateral shearing interferometry when acquiring interferograms in a single frame. At the same time, the PRSI method is proposed in a targeted manner by combining the characteristics of the simultaneous phase shift interferogram itself. The test results show that the high-resolution simultaneous phase-shifted interferogram reconstructed using this method outperforms the existing mainstream interpolation methods in terms of image metrics and phase reconstruction results. Still, the parameter setting of the interpolation method in this paper is based on the ideal simultaneous phase-shifted interferogram, which makes it impossible to achieve the optimal reconstruction effect in highly noisy scenes. Based on this, our subsequent work will mainly consider establishing adaptive parameter adjustment for the PRSI method in multiple scenes, to further expand the applicability and robustness of the method. Also, due to the similar interpolation principle, the method has the potential to be applied to other infrared or standard focal plane measurement systems to achieve high-resolution reconstruction after adjusting the appropriate parameter settings.

Funding

Basic Research (2020-JCJQ-JJ-429); Shaanxi Provincial Science and Technology Department (2022JM-345); Basic Research (JCKY2020426B009).

Acknowledgments

We thank the other members of the shear interference group at the Xi'an Technological University for their discussions and feedback on this work.

Disclosures

The authors declare no conflicts of interest.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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$w_{θ}$	horizontal	$45^{\circ}$ direction	$135^{\circ}$ direction	vertical
$I_{0}$	0.4951	0.4913	0.4876	0.4932
$I_{45}$	0.4951	0.5030	0.5050	0.4971
$I_{90}$	0.4990	0.5030	0.5110	0.5090
$I_{135}$	0.4951	0.4971	0.4990	0.4913
mean	0.4961	0.4986	0.5007	0.4977

Average PSNR (dB)	BL	BC	BS	PCDP	EARI	NP	PRSI
$I_{0}$	34.0424	34.0432	33.8104	34.4817	34.2338	34.5818	35.6076
$I_{45}$	33.6686	33.6693	33.4186	34.1685	34.1362	34.0846	35.2836
$I_{90}$	33.9920	33.9925	33.7377	34.6306	34.5740	34.5065	35.5937
$I_{135}$	33.6686	33.6704	33.4186	34.1802	33.9670	33.9146	35.0626
Average RMSE	BL	BC	BS	PCDP	EARI	NP	PRSI
$I_{0}$	0.0422	0.0420	0.0432	0.0391	0.0402	0.0388	0.0352
$I_{45}$	0.0474	0.0470	0.0487	0.0447	0.0457	0.0464	0.0393
$I_{90}$	0.0477	0.0475	0.0489	0.0443	0.0463	0.0482	0.0396
$I_{135}$	0.0474	0.0471	0.0487	0.0445	0.0451	0.0463	0.0404

Test Mirror Types	Method	BL	BC	BS	PCDP	EARI	NP	PRSI
Low-frequency dominant ( $c = 1 / 100$ , $e^{2} > 1$ )	ΔPV/λ	0.0014	0.0017	0.0029	0.0009	0.0013	0.0015	0.0004
Low-frequency dominant ( $c = 1 / 100$ , $e^{2} > 1$ )	ΔRMS/λ	0.0012	0.0013	0.0009	0.0004	0.0007	0.0011	0.0002
Mid-frequency dominant ( $c = 1 / 10 00$ , $e^{2} = 1$ )	ΔPV/λ	0.0027	0.0021	0.0023	0.0012	0.0016	0.0019	0.0006
Mid-frequency dominant ( $c = 1 / 10 00$ , $e^{2} = 1$ )	ΔRMS/λ	0.0018	0.0015	0.0016	0.0007	0.0009	0.0014	0.0003
High-frequency dominant ( $c = 1 / 10 00$ , $e^{2} = 1$ )	ΔPV/λ	0.0783	0.0775	0.0764	0.0743	0.0703	0.0738	0.0727
High-frequency dominant ( $c = 1 / 10 00$ , $e^{2} = 1$ )	ΔRMS/λ	0.1691	0.1680	0.1671	0.1648	0.1612	0.1645	0.1633

Test mirror	Concave reflector (caliber: 50.8 mm, focal length: 400 mm)
method	ZYGO	Untreat	BL	BC	BS	PCDP	EARI	NP	PRSI
PV/λ	0.3030	0.5898	0.5631	0.5564	0.5547	0.5387	0.5328	0.5436	0.5316
RMS/λ	0.0460	0.1986	0.1923	0.1901	0.1896	0.1735	0.1709	0.1758	0.1702
ΔPV/λ	-	0.2868	0.2601	0.2534	0.2517	0.2357	0.2298	0.2406	0.2286
ΔRMS/λ	-	0.1526	0.1463	0.1441	0.1436	0.1275	0.1249	0.1298	0.1242

$w_{θ}$	horizontal	$45^{\circ}$ direction	$135^{\circ}$ direction	vertical
$I_{0}$	0.4951	0.4913	0.4876	0.4932
$I_{45}$	0.4951	0.5030	0.5050	0.4971
$I_{90}$	0.4990	0.5030	0.5110	0.5090
$I_{135}$	0.4951	0.4971	0.4990	0.4913
mean	0.4961	0.4986	0.5007	0.4977

Improving the phase reconstruction accuracy of simultaneous phase-shifted lateral shearing interferometry using a polarization redundant sub-region interpolation method

Abstract

1. Introduction

2. Simultaneous phase-shifted lateral shearing interferometry

3. Polarization redundancy sub-region interpolation

3.1 Resolution reconstruction strategy

3.2 Polarization redundancy sub-regional approach

3.3 Optimizing edge weights

3.4 Complete interpolation reconstruction process

4. Results and discussion

4.1 Resolution reconstruction results

4.2 Simulated phase reconstruction results

4.3 Experimental phase reconstruction results

5. Conclusion

Funding

Acknowledgments

Disclosures

Data availability

References

Data availability

Cited By

Figures (10)

Tables (4)

Equations (21)

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