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Efficient acquisition of Mueller matrix via spatially modulated polarimetry at low light field

Tianlei Ning, Xiang Ma, Yuanhe Li, Yanqiu Li, and Ke Liu

Open Access Open Access

Abstract

Mueller polarimetry performed in low light field with high speed and accuracy is important for the diagnosis of living biological tissues. However, efficient acquisition of the Mueller matrix at low light field is challenging owing to the interference of background-noise. In this study, a spatially modulated Mueller polarimeter (SMMP) induced by a zero-order vortex quarter wave retarder is first presented to acquire the Mueller matrix rapidly using only four camera shots rather than 16 shots, as in the state of the art technique. In addition, a momentum gradient ascent algorithm is proposed to accelerate the reconstruction of the Mueller matrix. Subsequently, a novel adaptive hard thresholding filter combined with the spatial distribution characteristics of photons at different low light levels, in addition to a low-pass fast-Fourier-transform filter, is utilized to remove redundant background noise from raw-low intensity distributions. The experimental results illustrate that the proposed method is more robust to noise perturbation, and its precision is almost an order of magnitude higher than that of the classical dual-rotating retarder Mueller polarimetry at low light field.

© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement

1. Introduction

The realization of polarization measurement at low light field is of great significance in numerous applications [1–3]. It is necessary for the development of polarimetry with the ability to measure polarization information at low light field based on the following two scenarios. First, polarization measurement is implemented in the field with weak echo signals, such as remote sensing [4,5], quantum optics [6], and biomedical diagnosis [7]. The better performance of polarization measurement at low light field means that more abundant polarization information can be obtained. Second, weak light illumination is usually required because irradiation by strong light damages or changes the polarization properties of the sample, for example, in the measurement of living biological tissues [8,9] and micro-nano materials [3,10]. Hence polarimetry is often conducted in scenes with low SNR, resulting in that the development of polarimeters with the ability to measure polarization information at low light field is indispensable.

Mueller polarimetry [11,12] is capable of characterizing abundant polarization information including the diattenuation, retardation, and depolarization properties of a surface, optical element, or specimen. A method for Mueller polarimetry is often characterized by time-sequentially measuring the intensity change using rotating polarization elements [13–17]. The 16 Mueller matrix components can be calculated after measuring the intensity under at least 16 unique combinations of the polarization state generator (PSG) and polarization state analyzer (PSA). However, the method of rotating optical components is time-consuming, and it is only suitable for static or slowly varying processes. To reduce acquisition time, both polarized sensitive cameras [18] and 3 × 3 and 3 × 4 Mueller matrix polarimeter [19,20] were implemented, which successfully improve the measurement efficiency of Mueller matrix. Besides, time-varying retarders such as liquid crystal retarders [21–24] and photoelastic modulators [25] are also developed to accelerate the reconstruction of Mueller matrix. Each of these polarimeters has been shown to have certain advantages based on accuracy, speed, or compactness at strong light field. However, the above experimental schemes are easily disturbed by noise in the measurement caused by multiple exposures or division of focal plane, which may not be suitable for polarization measurement at extremely low light field [26,27]. Mueller polarimetry uses complex multi-channel systems that can be modulated by different optical elements in each channel [28–33]. Although such multi-channel schemes can measure the polarization properties with better simultaneity, this approach may reduce the signal-to-noise ratio (SNR) owing to the multiple beam splitters, and it is not suitable for polarization measurement at low light field. To overcome these drawbacks of polarization measurements for complex multi-channel systems or multiple exposures and division of focal plane, polarimetric measurement with simplified system configuration has been proposed to measure the Stokes vector in a strong light field. This is performed by using spatially modulated elements, including gradient index (GRIN) lens [34], birefringent polarizing elements [35,36].

Recently, Gao et al. presented a spatially modulated polarimeters based on zero-order vortex half-wave retarder (ZVHR), which has been proved to be accurate and robust, and can measure the state of completely polarized beams in a simple way [37–39]. However, the spatially modulated ellipsometric scheme based on ZVHR cannot achieve spatial modulation for the circular polarization component; hence, it is inaccurate for measuring partially polarized and unpolarized beams. Subsequently, Mueller-mapping star test polarimetry (MMSTP) was first used to verify the feasibility of measuring the Stokes vector at low light field in our work. Reportedly, a zero-order vortex quarter-wave retarder (ZVQR), instead of ZVHR, was used as a space-variant birefringence device to achieve spatial modulation for all polarization components. However, the calibration scheme of MMSTP is time-consuming as it requires measurement of the Mueller matrix of the ZVQR by 16 rotations of the polarization device [40]. To overcome these issues, optimized spatially modulated polarimetry (OSMP) with an efficient calibration method was proposed in our recent publication [41]. Essentially, the total polarimetric errors of all the optical elements, including the lens, were calibrated simultaneously by rotating the devices four times. Nevertheless, the hybrid gradient descent (HGD) method in OSMP is used to restore the polarization information, which causes back and forth oscillations near the optimal solution and hinders convergence efficiency. The existing Stokes-Mueller polarimetry performed in a strong light field has demonstrated the advantages of accuracy, high spatial resolution, and compactness. However, Mueller polarimetry with high robustness has not yet been demonstrated to calculate the Mueller matrix of a sample with high speed and accuracy in a low light field.

In this paper, to our best knowledge, the SMMP induced by a ZVQR, is for the first time applied to calculate the Mueller matrix of sample. The proposed method can achieve the fast acquisition for the Mueller matrix of sample only by four camera shots rather than 16 shots as in the state of the art. To further accelerate the calculation for the Mueller matrix, a momentum gradient ascent (MGA) algorithm is proposed. In addition, combined with the low pass fast-Fourier-transform filter, an adaptive hard thresholding filter is first presented to process the raw-low light image, which allows adaptively setting the threshold in front of the spatial distribution characteristics of photons at different low light levels. Based on the proposed method, a spatially modulated Mueller polarimeter with a collinear reflection structure (SMMP-CR) is designed to capture the polarization properties of thick biological tissues, and it exhibits excellent performance at low light fields.

The remainder of this paper is organized as follows. In Section 2, the theoretical model, calibration for the SMMP, and reconstruction method of the Mueller matrix are introduced. In Section 3, an experimental verification is presented to verify the accuracy of SMMP by measuring the standard devices in strong and low light fields. In Section 4, a polarization measurement for thick biological tissue is presented, and further discussion is provided. In Section 5, the conclusions and perspectives of this study are presented.

2. Acquisition of Mueller matrix via spatially modulated polarimetry

2.1. Analytical model of spatially modulated Mueller polarimeter

The polarization state of the incident light changes when it interacts with a sample. Without considering the nonlinear effects, the polarization states of the outgoing beam, are related to those of the input polarization states using the following expression [42].

$${\mathop s\limits^ \wedge}_{out} = M \cdot {\mathop s\limits^ \wedge}_{in},$$
where ${\mathop s\limits^ \wedge} _{in}$ and ${\mathop s\limits^ \wedge} _{\textrm{out}}$ represent the polarization states of the incoming and outgoing beams of the sample respectively, M is the $4 \times 4$ real Mueller matrix of the sample. Determination of the full Mueller matrix M using SMMP requires a polarization state generator (PSG) capable of generating light with at least four known linearly independent Stokes vectors ${\mathop s\limits^ \wedge} _{in}$, and a spatially modulated polarization state analyzer (SMPSA) that analyzes the four Stokes vectors ${\mathop s\limits^ \wedge} _{\textrm{out}}$ of the output light leaving the sample. The ${\mathop s\limits^ \wedge} _{\textrm{out}}$ is defined as follows:
$${\mathop {\textrm{s}}\limits^ \wedge} _{\textrm{out}} = \left[ {\begin{array}{{@{}cccc@{}}} \overrightarrow {\textrm{s}}_{\textrm{out}}^1&\overrightarrow {\textrm{s}}_{\textrm{out}}^2&\overrightarrow {\textrm{s}}_{\textrm{out}}^3&\overrightarrow {\textrm{s}}_{\textrm{out}}^4 \end{array}} \right] \cdot \left[ {\begin{array}{{@{}cccc@{}}} {\textrm{s}_0^1}&0&0&0\\ 0&{\textrm{s}_0^2}&0&0\\ 0&0&{\textrm{s}_0^3}&0\\ 0&0&0&{\textrm{s}_0^4} \end{array}} \right],$$
where $\overrightarrow s _{out}^i$ (with i = 1, 2, 3, 4) is a normalized Stokes vector of the outgoing light of the sample, and $s_0^i$ represents the total intensity of the i-th outgoing light of the sample. Here the normalized Stokes vector is written as $\overrightarrow s _{out}^i = {[\begin{array}{{@{}cccc@{}}} 1&{{s_1}}&{{s_2}}&{{s_3}} \end{array}]^T}$, where “T” denotes the transposition of the matrix. After the polarized light is characterized by four different Stokes vectors passing through the SMPSA, the four intensity distributions of the detection plane are given by
$$[\begin{array}{{@{}cccc@{}}} {{I_1}}&{{I_2}}&{{I_3}}&{{I_4}} \end{array}] = [1{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} 0{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} 0{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} 0] \cdot {M_{SMPSA}} \cdot {\mathop s\limits^ \wedge} _{\textrm{out}},$$
in which ${I_i}$ is the intensity distribution of the detection plane, and the Mueller matrix of SMPSA, ${M_{SMPSA}}$, is denoted as
$${M_{SMPSA}}\textrm{ = }\left[ {\begin{array}{{@{}cccc@{}}} {{m_{00}}}&{{m_{01}}}&{{m_{02}}}&{{m_{03}}}\\ {{m_{10}}}&{{m_{11}}}&{{m_{12}}}&{{m_{13}}}\\ {{m_{20}}}&{{m_{21}}}&{{m_{22}}}&{{m_{23}}}\\ {{m_{30}}}&{{m_{31}}}&{{m_{32}}}&{{m_{33}}} \end{array}} \right].$$

Then, Eq. (3) can be transferred as

$$[\begin{array}{{@{}cccc@{}}} {{I_1}}&{{I_2}}&{{I_3}}&{{I_4}} \end{array}] = [\begin{array}{{@{}cccc@{}}} {{m_{00}}}&{{m_{01}}}&{{m_{02}}}&{{m_{03}}} \end{array}] \cdot \left[ {\begin{array}{{@{}cccc@{}}} {\overrightarrow s_{out}^1}&{\overrightarrow s_{out}^2}&{\overrightarrow s_{out}^3}&{\overrightarrow s_{out}^4} \end{array}} \right] \cdot \left[ {\begin{array}{{@{}cccc@{}}} {s_0^1}&0&0&0\\ 0&{s_0^2}&0&0\\ 0&0&{s_0^3}&0\\ 0&0&0&{s_0^4} \end{array}} \right].$$

The theoretical model is been established in Eq. (5), which can encode the polarization information of the outgoing light of the sample into the intensity distribution of the imaging system. Thus, the normalized Stokes vector $\overrightarrow s _{out}^i$ can be constructed in section 2.3, provided that intensity distributions and Mueller matrix parameters of SMPSA in the Eq. (5) have been acquired. To reconstruct the full Stokes vectors, it is also necessary to calculate the total intensity $s_0^i$ of the outgoing light of the sample. If the parameters in Eq. (5) have been acquired, $s_0^i$ can be calculated by

$$s_0^i = \sum {{I_i}} /\sum {([\begin{array}{{@{}cccc@{}}} {{m_{00}}}&{{m_{01}}}&{{m_{02}}}&{{m_{03}}} \end{array}] \cdot \overrightarrow s _{out}^i)} ,$$
where $\sum $ is the summation notation that adds the value within effective pixels. Note that effective pixels are the area of intensity distributions that are used to calculate the polarization information by data processing. The Mueller matrix of the SMPSA changes from pixel to pixel over the image grabbed by the camera because of the addition of ZVQR, so the SMPSA should be calibrated pixel by pixel to achieve polarization measurement with high accuracy. The next subsection discusses how to obtain the Mueller matrix parameters of the SMPSA, which can be regarded as a process of calibration for the SMPSA.

2.2. Calibration for spatially modulated Mueller polarimeter

To measure ${\overrightarrow s _{out}}$ with high accuracy, the total polarimetric errors caused by the alignment and processing errors of the SMPSA devices must be calibrated. According to Eq. (5), four intensity distributions of known polarization states are required to calibrate the four model parameters ${m_{00}}$, ${m_{01}}$, ${m_{02}}$, ${m_{03}}$. For OSMP proposed by us [41], the intensity distribution of the horizontal polarization, 45° linear polarization, as well as right and left circular polarization, by rotating the both polarizer and wave plate, were selected to calibrate the model parameters. However, owing to the requirement of rotating multiple devices, such a calibration scheme is not conducive to the stability of the system. Here, we present a simplified calibration method by rotating the wave plate of the PSG four times without rotating the polarizer to calibrate the total polarimetric errors of the SMPSA. This procedure uses only air as a reference sample, and relies on ready-to-use cameras. When air is measured as a reference sample, four groups of linearly independent Stokes vectors are generated by rotating the wave plate of the PSG four times, and the four intensity distributions are obtained in the detection plane after the outgoing light of the sample is modulated by the SMPSA. The optimal four rotation angles of wave plate can be determined by optimizing the minimum condition number [43], in order to reduce the sensitivity to various error sources. The four rotation angles of wave plate of the PSG were set to ±15° and ±52 °in this paper. The four Stokes vectors generated by the PSG can be written as ${\mathop s\limits^ \wedge} _{in} = [{\overrightarrow s _1}{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\overrightarrow s _2}{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\overrightarrow s _3}{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\overrightarrow s _4}]$. The four intensity distributions used for the calibration can be formulated as

$$[I_\textrm{1}^{air}{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} I_2^{air}{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} I_3^{air}{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} I_4^{air}] = [\begin{array}{{@{}cccc@{}}} {{m_{00}}}&{{m_{01}}}&{{m_{02}}}&{{m_{03}}} \end{array}] \cdot {M_{air}} \cdot [{\overrightarrow s _1}{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\overrightarrow s _2}{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\overrightarrow s _3}{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\overrightarrow s _4}],$$
where ${M_{air}}$ represents the Mueller matrix of air, $I_i^{air}$ is the intensity distribution corresponding to the i-th input polarization state ${\overrightarrow s _i}$(i = 1,2,3,4). Considering the ${M_{air}}$ as an identity matrix, the Eq. (7) can be written as
$$[\begin{array}{{@{}cccc@{}}} {{m_{00}}}&{{m_{01}}}&{{m_{02}}}&{{m_{03}}} \end{array}] = [I_\textrm{1}^{air}{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} I_2^{air}{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} I_3^{air}{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} I_4^{air}] \cdot {[{\overrightarrow s _1}{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\overrightarrow s _2}{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\overrightarrow s _3}{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\overrightarrow s _4}]^{ - 1}}.$$

Furthermore, the theoretical model was calibrated by applying the measured $I_\textrm{1}^{air},I_2^{air},I_3^{air},I_4^{air}$ to Eq. (8), by considering the total polarimetric error of SMPSA.

2.3. Reconstruction of Mueller matrix

This subsection presents the reconstruction for Mueller matrix of sample, including how to establish the objective function and search for the optimal solution for the polarization state of the outgoing light of the sample. A quantitative link among the intensity distribution in the detection plane, the polarization states of the outgoing light of the sample, as well as the Mueller parameters of the SMPSA was developed. This model provides a polarization signature that the intensity distribution modulated by SMPSA can be uniquely tied to the output polarization state of the sample. Hence, the output polarization state of the sample could be restored from an intensity distribution by one camera shot. Inspired from image quality assessment [44], a normalized structural SIMilarity (NSSIM) index, was used to precisely evaluate the similarity of the intensity distribution measured in the experiment and theoretical intensity distributions of different polarization states according to the spatial characteristics of photons. The polarization state to be measured can be restored by solving the optimization problem which is to find the maximal similarity between theoretical intensity distributions of specific polarization states (${\overrightarrow s _{\max }}$) and measured intensity distribution. Then, the polarization state to be measured is endowed with ${\overrightarrow s _{\max }}$. The measured intensity distribution in the experiment and the theoretical intensity distribution corresponding to arbitrary polarization states are denoted as O and I, respectively. To extract the spatial characteristics of photons of the detected intensity distribution, intensity distributions O and I must be normalized as

$${O_\textrm{i}} = \frac{{{O_\textrm{i}}}}{{\sum\limits_{\textrm{i = 1}}^N {{O_\textrm{i}}} }}{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} ,{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {I_\textrm{i}} = \frac{{{I_\textrm{i}}}}{{\sum\limits_{\textrm{i = 1}}^N {{I_\textrm{i}}} }},$$
where N is the total number of pixels. The NSSIM index [44] is defined as,
$$NSSIM(O,I) = {(l(O,I))^\alpha } \cdot {(c(O,I))^\beta } \cdot {(s(O,I))^\gamma },$$
where $NSSIM(O,I)$ represents the similarity of intensity distributions O and I; and, $l(O,I)$, $c(O,I)$, and $s(O,I)$ are the luminance comparison function, contrast comparison function, structure comparison function respectively, where α > 0, β > 0 and γ > 0 are the parameters used to adjust the relative importance of the three components, and we set α=β=γ=1. The luminance comparison function $l(O,I)$[44] is formulated as
$$l(O,I) = \frac{{2{\mu _O}{\mu _I}}}{{\mu _O^2 + \mu _I^2}},$$
where ${\mu _O}$ and ${\mu _I}$ are the mean values of the intensity distributions O and I. The contrast comparison function $c(O,I)$[44] is defined as
$$c(O,I) = \frac{{2{\sigma _O}{\sigma _I}}}{{\sigma _O^2 + \sigma _I^2}},$$
where ${\sigma _O}$ and ${\sigma _I}$ are the standard deviation of the intensity distributions O and I, respectively. The structure comparison function $s(O,I)$[44] is written as
$$s(O,I) = \frac{{2{\sigma _{OI}}}}{{{\sigma _O}{\sigma _I}}},$$
where ${\sigma _{OI}}$ is the covariance of intensity distribution O and I. Thus, the NSSIM index [44] can be obtained as follows
$$NSSIM(O,I) = \frac{{6{\mu _O}{\mu _I}{\sigma _{OI}}}}{{(\mu _O^2 + \mu _I^2)(\sigma _O^2 + \sigma _I^2)}}.$$

Therefore, the objective function is established to reconstruct the polarization state of the outgoing light of the sample, which can be formulated as

$$\begin{aligned} {\overrightarrow s _{\max }} &= \mathop {\arg \max }\limits_{{{\overrightarrow s }_{out}}} (NSSIM(O,I))\\ &= \mathop {\arg \max }\limits_{{{\overrightarrow s }_{out}}} F(\overrightarrow s ). \end{aligned}$$
where $F(\overrightarrow s )$ is the cost function of the normalized Stokes vector $\overrightarrow s$. Hence, the polarization state to be measured can be restored by solving the optimization problem after acquiring the intensity distribution O. To further accelerate the calculation for ${\overrightarrow s _{\max }}$, the MGA algorithm, learning from the gradient algorithm [45], is proposed to search for the optimal solution of Eq. (15) using iterative optimization, as shown in Fig. 1.

 figure: Fig. 1.

Fig. 1. Flowchart for the proposed MGA algorithm, and $|{\kern 1pt} {\kern 1pt} \cdot {\kern 1pt} {\kern 1pt} |$ is the absolute value function, and the number of iterations is represented as ${k_{\max }}$.

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In Fig. 1, the following terms are used:

$$F_{{s_1}}^k = F(s_1^k,s_2^{k - 1},s_3^{k - 1}),$$
$$F_{{s_2}}^k = F(s_1^{k - 1},s_2^k,s_3^{k - 1}),$$
$$F_{{s_3}}^k = F_{{s_1}}^k(s_1^{k - 1},s_2^{k - 1},s_3^k),$$
where $F_{{s_i}}^k$ is the cost function corresponding to the Stokes parameter ${s_i}$ in the k-th iteration, and $s_i^k$ represents the i-th Stokes parameter in the k-th iteration. Then, the Stokes parameters $s_i^k$ in the k-th iteration are updated to
$$s_i^k = s_i^{k - 1} + d_{{s_i}}^k,$$
where $d_{{s_i}}^k$ is the search factor in the k-th iteration corresponding to the $s_i^k$. To weaken the oscillation back and forth near the optimal solution and expedite convergence, the MGA algorithm, whose search factor $d_{{s_i}}^k$ is not only related to the gradient of the cost function but also depends on the search factor in the previous iteration, was proposed. So we have
$$d_{{s_i}}^k = g_{{s_i}}^k \cdot N_{{s_i}}^k + d_{{s_i}}^{k - 1} \cdot M_{{s_i}}^k,$$
where $g_{{s_i}}^k$ is the gradient of the cost function; and, $N_{{s_i}}^k$ and $M_{{s_i}}^k$ are the gradient contraction and momentum factors. $g_{{s_i}}^k$ can be written as
$$g_{{s_i}}^k = \frac{{F_{{s_i}}^k - F_{{s_i}}^{k - 1}}}{{s_i^k - s_i^{k - 1}}}.$$

Here, $N_{{s_i}}^k$ and $M_{{s_i}}^k$ can be calculated as following

$$N_{{s_i}}^k = {{t_{{s_i}}^k \cdot sign(g_{{s_i}}^k)} / {g_{{s_i}}^k}},$$
$$M_{{s_i}}^k = {{r_{{s_i}}^k \cdot sign(d_{{s_i}}^{k - 1})} / {d_{{s_i}}^{k - 1}}},$$
where $t_{{s_i}}^k$ and $r_{{s_i}}^k$ are the iterative step lengths, and sign represents a symbolic function. If x < 0, sign(x)=-1; if x$\ge $0, sign(x) = 1. The MGA algorithm updates the $t_{{s_i}}^k$ and $r_{{s_i}}^k$ to a smaller value according to the change in the gradient direction in the iterative process. In Eq. (20), the search factor $d_{{s_i}}^k$ of the Stokes parameters depends on the search factor $d_{{s_i}}^{k - 1}$ that can be regarded as an inertia factor. The inertia implies that it will help accelerate ascent in the initial stage of search, and slow down the ascent when the gradient turns in the opposite direction. Essentially, this method can accelerate convergence and weaken the oscillation back and forth near the optimal solution. Thus, the polarization state of the outgoing light of the sample can be restored by solving the optimization problem using the MGA algorithm. Then, the $s_0^i$ can be calculated using Eq. (6). Similarly, the Mueller matrix of the sample in Eq. (1) can be acquired after complete reconstruction of the four Stokes vectors of the outgoing light of sample. Thus, the following is obtained.
$$M = {\mathop s\limits^ \wedge} _{in} \cdot inv({\mathop s\limits^ \wedge} _{in}),$$
where inv represents the inversion of the matrix. ${\mathop s\limits^ \wedge} _{in}$ can be calculated after the four Stokes vectors of the outgoing light of the sample have been acquired by solving the optimization problem in Eq. (15).

To verify the convergence efficiency of the MGA, the Mueller matrix of air was measured using the SMMP in a strong light field, and the maximum error (ME) of the Mueller matrix elements was calculated as a standard for optimization effectiveness. Compared with the HGD algorithm in our previous work, the proposed MGA algorithm can significantly speed up convergence, as shown in Fig. 2.

 figure: Fig. 2.

Fig. 2. The convergence curves of different optimization algorithm.

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3. Experimental verification

Quantitative experiments are presented in this section to validate the accuracy and feasibility of the proposed method. The four types of samples known as the Mueller matrix, including air and quarter wave plate (QWP) with the fast-axis azimuthal angle at 0, $\pi /4$, and $\pi /2$ are typically used to verify the measurement accuracy. Among all polarimeters, DRRP has been validated because of its advantages of accuracy and robustness of noise. Herein, the Mueller matrix of the sample was measured using the proposed SMMP and the measured results were compared with those of the classical DRRP at strong and low light fields. For a measurement of Mueller matrix by SMMP, it takes about 20s to collect data in the experimental process including the 4 rotations of wave plate and camera shots. Benefiting from the acceleration of MGA method, the data processing only needs ∼50 ms to get the Mueller matrix of sample to be measured. All computations were implemented by MATLAB and carried out on a computer with Intel R CoreTMi5-8400 CPU, 2.8 GHz, and 8 GB of RAM.

3.1. Measurement of Mueller matrix performed at strong light field

To verify the proposed SMMP experimentally, the setup shown schematically in Fig. 3 was used. The SMMP consists of an LED source (M625L4, Thorlabs, with center wavelength = 625 nm) that is coupled into a fiber by a doublet lens1 and then collimated by another doublet lens2 into free space. It also includes a PSG, SMPSA, and light intensity detector sCMOS (produced by ANDOR, Zyla - 4.2 PLUS). This collimated illumination light was modulated by a PSG, which consisted of a fixed polarizer P1 (10LP-VISB, Newport) and rotating QWP Q1 (AQWP10M-580, Thorlabs) driven by a computer-controlled servo motor driver (PI U-651.03). Four groups of linearly independent polarization lights modulated by the PSG evenly irradiated the sample to be measured. For the typical DRRP configuration, the polarization state analyzer (PSA) consists of the same type of fixed polarizer P2 and rotating quarter-wave plate Q2 as those in the PSG. Unlike DRRP, SMMP adds a ZVQR (WPV10-633-QUARTER-SP, Thorlabs) in front of the PSA to achieve spatial modulation for all polarization components of the outgoing light of the sample. ZVQR is a zero-order vortex quarter-wave retarder, and its fast-axis azimuth rotates continuously over the area of the optics about the center [46,47]. The fast axis azimuth varies linearly with the polar angle as follows:

$$\xi \textrm{ = }\frac{\textrm{m}}{2}\varphi \textrm{ + }\delta ,$$
where m equals to 2, $\xi $ is the fast-axis orientation, given the azimuth $\varphi$ and $\delta $ is the fast axis orientation when $\varphi$ equals to zero. When the light propagated past the SMPSA system, its resulting transverse irradiance distribution provided a signature that was unambiguously linked to the polarization state of the outgoing light of sample. Through this method, we can reconstruct the Stokes vector of outgoing light of sample by one camera shot rather than four shots like the DRRP. Therefore, the Mueller matrix of the sample could be calculated using four intensity distributions photographed four times by the scientific complementary metal-oxide-semiconductor (sCMOS).

 figure: Fig. 3.

Fig. 3. Schematic of the experimental system of the spatially modulated Mueller polarimeter (SMMP). It included a collimated LED, polarization state generator (PSG), spatially modulated polarization state analyzer (SMPSA), linear horizontal polarizers P1 and P2, quarter-wave plates Q1 and Q2, zero-order vortex quarter wave retarder (ZVQR), focusing lens (Lens3), and scientific complementary metal-oxide-semiconductor (sCMOS).

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The alignment and processing errors of the devices must be calibrated before the comparative experiment of measuring the Mueller matrix using the SMMP and DRRP. The detailed calibration process can be found in our previous study [48]. The Mueller matrix of air was measured using the DRRP to verify the effect of calibration, and the maximum measurement error of air after calibration was approximately 0.6%. In addition, to calibrate the total polarimetric errors of the SMPSA, four intensity distributions in the experiment were acquired by rotating the wave plate of the PSG four times and calculating the actual model parameters ${m_{00}},{\kern 1pt} {m_{01}},{m_{02}},{\kern 1pt} {m_{03}}$ in Eq. (8), as shown in Fig. 4.

 figure: Fig. 4.

Fig. 4. The calibrated model parameters, (a): ${m_{00}}$, (b): ${m_{01}}$, (c): ${m_{02}}$, (d): ${m_{03}}$. The calculated pixel area is 80 × 80, and a circular mask is added to select the effective pixel area.

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In the following experiments, the performance of the proposed method was tested with air and the QWP with the fast-axis azimuthal angle oriented at 0°, 45°, and 90° with respect to the x-axis such that four different Mueller matrices were measured. The QWPs with the fast-axis azimuthal angle oriented at 0°, 45°, and 90° are written as QWP (0°), QWP (45°) and QWP (90°) in this paper. The Mueller matrix of the sample was calculated using the proposed SMMP after four intensity distributions were acquired by rotating the wave plate of the PSG at ±15° and ±52°. Taking air and QWP (0°) as an example, the four measured intensity distributions are shown in Fig. 5 and Fig. 7, and the four theoretical intensity distributions are shown in Fig. 6 and Fig. 8.

 figure: Fig. 5.

Fig. 5. The measured four intensity distributions with air as tested sample, (a): -15°, (b): -52°, (c): 52°, (d):15°.

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 figure: Fig. 6.

Fig. 6. The theoretical four intensity distributions with air as sample, (a): -15°, (b): -52°, (c): 52°, (d):15°.

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 figure: Fig. 7.

Fig. 7. The measured four intensity distributions with QWP (0°) as tested sample, (a): -15°, (b): -52°, (c): 52°, (d):15°.

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 figure: Fig. 8.

Fig. 8. The theoretical four intensity distributions with QWP (0°) as sample, (a): -15°, (b): -52°, (c): 52°, (d):15°.

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The Mueller matrices are measured using the SMMP, and the absolute errors of the Mueller matrix elements summarize the measured results for all the samples, as shown in Table 1. The experimental results of the mean value of the Mueller matrix elements along with the difference from the theoretical values are reported herein. Root mean square (RMS) is regard as a reliable index to verify the precision and accuracy of the results [49]. Besides, the ME of the Mueller matrix elements can scientifically weigh the measurement accuracy of the Mueller polarimeter. Hence, these two indexes are used to evaluate the accuracy of polarization measurement in this paper. As shown in Table 1 and 2, the RMS and ME of Mueller matrix of sample measured by the SMMP at strong light field were smaller than 0.03 and 0.06. The measurement accuracy of the Mueller matrix for different samples was different, which may be due to the following reasons. First, the measurements of the Mueller matrix for different samples were nonsimultaneous, and the vibration of the light source and instrument changed when different samples were measured. In addition, the determination of the fast axis of a measured QWP did not strictly agree with the theoretical angles at 0°, 45°, and 90° due to the intrinsic process error. To maintain the same experimental conditions to the greatest extent, the DRRP (removing the ZVQR in the SMMP) was also used to measure the above sample, and the measured mean value of the Mueller images is shown in Table 3. The Mueller images of the four samples using the DRRP, as shown in Fig. 9. As shown in Table 3 and 4, the RMS and ME of Mueller matrix of sample measured by the DRRP were smaller than 0.02 and 0.05, and its measurement accuracy was slightly higher than that of the SMMP in strong light field. The proposed SMMP obtains the numerical solution of the Mueller matrix by analyzing the spatial distribution characteristics of photons of the theoretical and measured distributions. The measured result of the Mueller matrix using the SMMP constantly approaches the optimal solution; however, achieving consistency with the theoretical value is difficult. The DRRP can obtain an analytic solution of the Mueller matrix of the sample on the basis of the intensity variation relationship of acquiring the 16 intensity distributions. Such an intensity variation relationship is negligibly affected by noise in a strong light field, and thus the measurement accuracy of the DRRP is slightly higher than that of the SMMP in strong light fields.

 figure: Fig. 9.

Fig. 9. The Mueller matrix images of four sample measured by DRRP. (a): air, (b): QWP (0°), (c): QWP (45°), (d): QWP (90°).

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Table 1. Mean values and absolute error of the Mueller matrix of samples measured by SMMP at strong light field

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Table 2. RMS and ME of the Mueller matrix of samples measured by SMMP at strong light field

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Table 3. Mean values and absolute error of the Mueller matrix of samples measured by DRRP at strong light field

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Table 4. RMS and ME of the Mueller matrix of samples measured by DRRP at strong light field

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3.2. Measurement of Mueller matrix performed at low light field

This section presents an overall experiment to verify the performance of the proposed method in the measurement of the Mueller matrix at low light field. As stated by the Society of Photo-Optical Instrumentation Engineers (SPIE), the definition of low light is that the local SNR (LSNR) is less than 5 [50]. The LSNR can be calculated as follows [50].

$$LSNR\textrm{ = }10\cdot \log _{10}^{({E_r} - {E_B})/{\sigma _B}},$$
where ${E_\textrm{r}}$ represents the mean value of the target area, ${E_B}$ and ${\sigma _\textrm{B}}$ are the mean value and standard deviation of the background noise, respectively.

Note that the intensity distribution at a low light field is almost submerged in the background noise. Noise is inevitably introduced in imaging systems and may severely damage the quality of the acquired images, thus resulting in poor accuracy of the Mueller matrix at low light fields. In this study, an adaptive hard thresholding filter (AHTF) that considers the spatial distribution characteristics of photons at different low light levels, in addition to a low-pass fast-Fourier-transform (LFFT) filter [51], was developed to remove redundant background noise from raw low light images. The overall denoising algorithm comprised two stages, as shown in Fig. 10, and was interpreted as follows.

 figure: Fig. 10.

Fig. 10. Flow chart of the denoising process.

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A. An LFFT filter was used to process the raw intensity distribution.

B. Next, the AHTF was applied to process the image from step A and obtain the final denoised image.

The spatial domain signal O must be transformed to the frequency domain written as follows:

$$F(u,v) = \sum\limits_{x = 1}^M {\sum\limits_{y = 1}^N {O(x,y)} } {e^{ - i2\pi (ux/M + vy/N)}},$$
where x and y denote the horizontal and vertical coordinates of the original image; M and N define the row height and column width of the original image (M = N = 80 in this paper), $O(x,y)$ indicates the gray value at coordinates x, y, and $F(u,v)$ is the value in the frequency domain. In this study, the LFFT filter was adopted because the noise herein was a high-frequency signal. The filter template was designed to filter out the high-frequency noise. The distance between the position $u,v$ and the center point is calculated as
$$D(u,v) = \sqrt {{{(u - M/2)}^2} + {{(v - N/2)}^2}} ,$$
where $D(u,v)$ is the distance. Further, the filter template is designed as
$$H(u,v) = \left\{ {\begin{array}{{c}} {1,{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} D(u,v) \le {d_0}}\\ {0,{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} D(u,v) > {d_0}} \end{array}} \right.{\kern 1pt} ,$$
where ${d_0}$ is the frequency threshold determined according to the different noise environment. The frequency spectrum can be expressed as
$$F(u,v) = F(u,v) \cdot H(u,v).$$

Through this method, an effective signal with a low frequency was preserved. Next, the signal was restored to the spatial domain using the inverse Fourier transform, as follows.

$$O(x,y) = \frac{1}{{MN}}\sum\limits_{x = 1}^M {\sum\limits_{y = 1}^N {F(u,v)} } {e^{i2\pi (ux/M + vy/N)}}.$$

Furthermore, the AHTF was applied to process the image denoised by the LFFT. For the reconstruction of Mueller matrix using the SMMP, the four intensity distributions acquired by rotating the wave plate of the PSG four times to ±15° and ±52° were placed in different LSNRs as shown in Fig. 11.

 figure: Fig. 11.

Fig. 11. The LSNR of four intensity distributions of different samples.

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Therefore, the same hard threshold (HT) can be applied to process the four intensity distributions with different LSNRs. In this study, the AHTF can be formulated as:

$$O(x,y) = O(x,y) - \frac{1}{{MN}}\sum\limits_{x = 1}^M {\sum\limits_{y = 1}^N {O(x,y)} } ,$$
in which $O(x,y)$ is the image denoised by LFFT.
$$O(x,y) = \left\{ {\begin{array}{{c}} {0,{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} O(x,y) < \gamma (LSNR)}\\ {{\kern 1pt} O(x,y) - \gamma (LSNR){\kern 1pt} {\kern 1pt} ,{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} O(x,y) \ge \gamma (LSNR){\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} } \end{array}} \right.{\kern 1pt} ,$$
where $\gamma (LSNR)$ indicates the adaptive HT, which can be determined according to the LSNR of different intensity distributions. Determining a reasonable $\gamma (LSNR)$ is the key to the AHTF. In this study, the calibrated model parameters in a strong light field were used to produce 21 non-noise intensity distributions of known polarization states by evenly rotating the wave plate from −50° to 50°, with a step of 5° in the simulation. Furthermore, the 21 non-noise intensity distributions were polluted by Gaussian noise. The difference between the LSNR of the intensity distribution polluted by Gaussian noise and that LSNR of intensity distribution measured by experiment at low light field is less than 0.5, in order to ensure that both intensity distributions are proximate. The SMPSA was used to measure the polarization states of the 21 intensity distributions polluted by noise, which were denoised using LFFT and an HT filter with different $\gamma$. The optimal $\mathop \gamma \limits^ \wedge $ can be determined
$$\mathop \gamma \limits^ \wedge{=} \mathop {\arg \min }\limits_\gamma {\kern 1pt} {\kern 1pt} ({\overrightarrow s _M} - {\overrightarrow s _T}),$$
where ${\overrightarrow s _M}$ and ${\overrightarrow s _T}$ are the measured and theoretical polarization states, respectively, which can be calculated by minimizing the error of the polarization states of the measured and theoretical results. Subsequently, the correlation curve of the LSNR and optimal HT could be fitted. Finally, the adaptive HT$\gamma (LSNR)$ was determined according to the fitting correlation curve after the raw intensity distribution was acquired in the experiment. Compared with the acquisition of fitting data from experimental measurement, such scheme to get the fitting data is conveniently and insensitive to system error, such as the angle error from the rotation of wave plate and drift of light source. The final denoised images shown in Fig. 12 and Fig. 13 were acquired after two denoising stages were performed.

 figure: Fig. 12.

Fig. 12. The measured four raw-intensity distributions with air as tested sample at low light field, (a1): -15°, (b1): -52°, (c1): 52°, (d1): 15°, and (a2)- (d2) are the final denoised images corresponding to (a1) - (d1).

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 figure: Fig. 13.

Fig. 13. The measured four raw-intensity distributions with QWP (0°) as tested sample at low light field, (a1): -15°, (b1): -52°, (c1): 52°, (d1): 15°, and (a2)- (d2) are the final denoised images corresponding to (a1) - (d1).

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The final denoised images are very close to the intensity distributions in a strong light field, as shown in Fig. 5 and Fig. 7. The Mueller matrices are measured by the SMMP and DRRP at low light fields, and the absolute errors of the Mueller elements summarize the measured results for all the samples as shown in Table 5 and 7. The RMS and ME of Mueller matrix of sample measured by the SMMP at low light field were smaller than 0.05 and 0.09 as shown in Table 6, which indicates that the SMMP combined with the AHTF algorithm can still measure the Mueller matrix with high accuracy at low light fields.

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Table 5. Mean values and absolute error of the Mueller matrix of samples measured by SMMP at low light field

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Table 6. RMS and ME of the Mueller matrix of samples measured by SMMP at low light field

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Table 7. Mean values and absolute error of the Mueller matrix of samples measured by DRRP at low light field

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Then, the Mueller matrices are measured by the DRRP at low light fields. The 16 raw-intensity distributions were detected using an sCMOS at a low light field, as shown in Fig. 14(a), before the Mueller matrix of air was reconstructed using the DRRP. Figure 14(b) shows the intensity distribution of the background noise, and the LSNRs of different intensity distributions are shown in Fig. 14(c), where the 16 intensity distributions are written as ${I_1},{I_2}, \cdots {I_{16}}$. In addition, Mueller images of four samples obtained by the DRRP at low light fields were denoised, as shown in Fig. 15. As shown in Table 7 and 8, the RMS and ME of Mueller matrix of sample measured by the DRRP were approximate to 0.3 and 0.8. and its measurement accuracy was extremely lower than that of the SMMP in a low light field. For the DRRP, evidently, the Mueller matrix elements were close to zero, and almost all polarization effects were missing at low light fields. Therefore, the measurement of the Mueller matrix with the DRRP was inaccurate for low light fields.

 figure: Fig. 14.

Fig. 14. (a) 16 raw-intensity distributions detected by the sCMOS; (b): intensity distribution of background noise; (c) LSNR of different intensity distributions ${I_1},{I_2}, \cdots {I_{16}}$.

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 figure: Fig. 15.

Fig. 15. Denoised Mueller images of four samples measured by the DRRP at low light field. (a): air, (b): QWP (0°), (c): QWP (45°), (d): QWP (90°).

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Table 8. RMS and ME of the Mueller matrix of samples measured by DRRP at low light field

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In this section, the effectiveness of the proposed method in the measurement of the Mueller matrix of a standard sample in a low light field was verified. Evidently, the SMMP exhibited better performance compared with the DRRP at a low light field, which can be attributed to the following reasons. For the DRRP, the analytic solution of Mueller matrix can be calculated based on the intensity variation relationship of the 16 intensity distributions acquired, the weakest of which is submerged in the background noise at low light fields, as shown in Fig. 14(a). The minimum value of LSNR is less than -5 in Fig. 14(c). Therefore, accurately calculating the Mueller matrix in analytical form according to the intensity variation relationship is challenging due to the interference of intense noise at low light fields, even if the intensity distributions have been denoised. In general, such a polarimetric scheme measures the Mueller matrix according to the variation relationship of overall intensity values of multiple images, which is susceptible to the influence of background noise at low light field [52]. For the proposed SMMP, the polarization information of the outgoing light of the sample is exactly mapped to a space-variant intensity distribution benefiting from the spatial modulation of the SMPSA. The numerical solution of the polarization state can be calculated by analyzing the spatial distribution characteristics of photons according to the Eq. (15). Therefore, the SMMP emphasizes the information gained from the shape of the intensity distribution instead of considering the overall intensity, which is not easily influenced by the power fluctuations of the light source [53]. Although the measured intensity distribution had a low LSNR, the shape of the intensity distribution could still be approximately maintained by the AHTF combined with an LFFT filter, as shown in Figs. 12 and 13 (a2) - (d2). Moreover, the NSSIM index, considering the characteristics of the spatial distribution of photons, has been used to reconstruct the polarization state of the outgoing light of the sample, which is conducive to weakening the impact of the energy fluctuation of the light source at low light fields. Consequently, the proposed method is rarely influenced by the power fluctuations of the light source and can open up new avenues for the measurement of the Mueller matrix at low light fields.

4. Polarization measurement for thick biological tissue

In the previous section, we have checked the validity of SMMP by measuring the standard sample as air and the QWPs. Furtherly, we will apply the SMMP to assess the sample whose Mueller matrix is unknown beforehand. A potential application of SMMP is to quickly capture the polarization properties of isotropic sample or the mean values of polarization properties of sample in the small area at low light field. The mean values of polarization properties of sample are usually utilized to realize the quantitative measurement and characterization of biological tissues [54,55]. In this section, we will assess the polarization properties of thick biological tissues.

4.1. Experimental system

For a clear comparison, two sets of polarization measurement systems were built based on a collinear reflection structure: a Mueller matrix polarimeter based on dual rotating retarders (DRRP-CR) and a spatially modulated Mueller polarimeter (SMMP-CR), as shown in Fig. 16.

 figure: Fig. 16.

Fig. 16. Photographs (a) and schematic diagram (b) of the instrument. The same devices in the two figures are outlined by dotted lines of the same color.

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The output light was produced by an LED source (Thorlabs MCWHLP1) with a wavelength of 400–700 nm, and a filter (Thorlabs FL632.8-1) was applied after the LED. Koehler illumination was used in the illumination module to ensure the uniformity of the illumination of the sample and to match the numerical aperture (NA) of the objective lens. The key structure of Koehler illumination was designed, as described in our previous publication [56]. This illumination light was modulated by a PSG, which consisted of a fixed polarizer P1 (Newport 10LP-VIS-B) and a rotating quarter-wave plate R1 (Thorlabs AQWP10M-580) driven by a computer-controlled servo motor driver (PI U-651.03). Three 50:50 nonpolarizing beam splitter cubes (NPBS) were used in the overall system, including NPBS1 (Thorlabs BSW26Rv), NPBS2 (ThorlabsBSW16), and NPBS3 (Thorlabs CCM1-BS013/M). The PSG-modulated illumination beam was first reflected by the beam splitter NPBS1, transmitted through the objective (Nikon, 10x), and then backscattered by the sample. The backscattered light was recollected by the same objective and transmitted through the NPBS1 again. A pinhole (diameter of 0.8 mm) was punched on the oxidized and blackened iron plate to limit the view of the sample to a small area. The SMMP-CR and DRRP-CR have a common illumination module, PSG, and objective. In the polarization analysis, the SMMP-CR used the SMPSA, whereas the DRRP-CR only used the PSA, according to the configuration of the dual-rotating quarter-wave plate. Compared with DRRP-CR, a ZVQR is added in the SMMP-CR. The pluggable ZVQR is placed on a mechanical rack (Thorlabs, LCP33/M) fixed on cage system. Only ZVQR will be removed when the DRRP-CR is used to acquire Mueller matrix images. The mechanical parts and optic axis of system were not moved during the conversion between SMMP-CR and DRRP-CR. The average reflectance and beam deviation of ZVQR are less than 0.5% and 20arcmin, respectively, according to the specifications from official website of Thorlabs. The high solidity of cage system and excellent flatness of mechanical rack ensure the stability in the process of measurement. In addition, a qCMOS (Hamamatsu, ORCA-Quest qCMOS) and sCMOS (Andor Zyla 4.2 Plus) were selected to detect the signals from the SMMP-CR and DRRP-CR, respectively. A reflector1 (Thorlabs BBE1-E02) was used to change the direction of light to ensure that the qCMOS could conveniently acquire the beam. For the DRRP-CR, a tube lens was used to match the objective lens to acquire clear images from the light backscattered by the sample. For the SMMP-CR, the intensity distribution after modulation by the SMPSA carried the polarization information of the outgoing light of the sample, which was focused on the qCMOS by Lens1, whose front focal plane is the position of the polarizer of the SMPSA. Under the same exposure time, the readout noise of the CMOS and the detected light intensity mainly determine the SNR of the detection signal at a low light field. Note that the qCMOS delivered an ultra-low readout noise of 0.27 electrons rms, which was nearly four times lower than the readout noise of 1 electrons rms of the sCMOS. To compare the performance of the DRRP-CR and SMMP-CR at low light fields fairly, the NPBS3 and reflector2 were used to degrade the light intensity entering the qCMOS, which was four times lower than the intensity detected by sCMOS.

4.2. Calibration of the DRRP-CR and SMMP-CR

Unlike the transmission Mueller matrix microscope, the polarimeter of the collinear reflection structure presented more errors derived from the NPBS1, objective lens, and pinhole used in this study, including the parasitic polarization artifacts introduced by the NPBS1 and the surface reflecting light from the NPBS1, objective lens, and pinhole without illuminating the sample. Hence, these adverse effects must be eliminated to achieve high-precision polarization measurements. Herein, the double-pass eigenvalue calibration method (double-pass ECM) [57,58] was adopted and previous research was referred to calibrate the DRRP-CR and PSG of the SMMP-CR. During calibration, a mirror and three reference samples, including polarizers and a wave plate, were measured to determine the instrument matrices of the PSG and PSA. Owing to the involvement of the ZVQR, the SMPSA of the SMMP-CR was calibrated using the method proposed in Section 2.2. A mirror was selected as a reference sample to calibrate the total polarimetric errors of the SMPSA in the SMM-CR. Specular light without illuminating the sample should be eliminated to achieve high-precision polarization measurements. Therefore, the calibrated model in Eq. (8) can be formulated as follows.

$$[\begin{array}{{@{}cccc@{}}} {{m_{00}}}&{{m_{01}}}&{{m_{02}}}&{{m_{03}}} \end{array}] = [I_1^{mirror} - {r_1}{\kern 1pt} I_2^{mirror} - {r_2}{\kern 1pt} I_3^{mirror} - {r_3} {\kern 1pt} I_4^{mirror} - {r_4}] \cdot {[{\overrightarrow s _1}{\kern 1pt} {\overrightarrow s _2}{\kern 1pt} {\overrightarrow s _3}{\kern 1pt} {\overrightarrow s _4}]^{ - 1}} \cdot M_{mirror}^{ - 1},$$
where ${r_i}$ is the intensity distribution generated by the specular light without illuminating the sample, and ${M_{mirror}}$ is the Mueller matrix of the mirror, $I_i^{mirror}$i s the intensity distribution corresponding to the i-th input polarization state ${\overrightarrow s _i}$(i = 1,2,3,4).

The calibrated instruments were used to measure the Mueller matrix of the mirror using DRRP-CR and SMMP-CR to verify the effect of calibration, and the mean values of each Mueller elements are listed in the Table 9.

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Table 9. The measured mean values of Mueller matrix of mirror after calibration

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4.3. Experimental results and discussion

The DRRP-CR and SMMP-CR were used to measure biological samples whose Mueller matrix was unknown beforehand in strong and low light fields. Pork and chicken breast tissues were selected to verify the reliability of the proposed method, as shown in Fig. 17.

 figure: Fig. 17.

Fig. 17. Photographs of the biological samples: (a) pork tissue, (b) chicken breast tissue.

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In the experiment, the pork and chicken breasts were cut into thick slices with a thickness of approximately 1 cm. In the measurement process, the SMMP-CR was first used to calculate the Mueller matrix of the sample at the strong light field; subsequently, the Mueller matrix was measured at a low light field by attenuating the electric current of the LED. Next, the DRRP-CR was used to measure the Mueller matrix of the sample at strong and low light fields by removing the ZVQR of the SMMP-CR. Figure 18 shows the normalized Mueller matrix images and total intensity images of pork and chicken breasts measured by the DRRP-CR under a strong light field. The field range of measured data is about 200µm × 200µm.

 figure: Fig. 18.

Fig. 18. (a) and (b) are normalized Mueller matrix images of the biological samples; (c) and (d) are the total intensity images of the biological samples; (a) and (c): pork tissue; (b) and (d): chicken breast tissue.

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In the same position of the tissue, the SMMP-CR was used to capture the overall polarization property of the sample in a small area delineated by the pinhole. Four intensity distributions were acquired by rotating the wave plate of the PSG at ±15° and ±52°. Figures 19 and 20 show the four intensity distributions acquired by measuring the pork and chicken breast tissues in a strong light field, respectively.

 figure: Fig. 19.

Fig. 19. Four intensity distributions acquired with pork tissue as the tested sample, (a) −15°, (b) −52°, (c) 52°, (d) 15°.

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 figure: Fig. 20.

Fig. 20. Four intensity distributions acquired with chicken breast tissue as the tested sample, (a) −15°, (b) −52°, (c) 52°, (d) 15°.

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The mean values of the Mueller matrices of the two types of samples were measured using the DRRP-CR and SMMP-CR under a strong light field, as presented in Table 10 and Table 11.

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Table 10. The mean value of Mueller matrix of samples measured by DRRP-CR at strong light field

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Table 11. The mean value of Mueller matrix of samples measured by SMMP-CR at strong light field

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The performance of the DRRP-CR and SMMP-CR in strong light fields was tested to capture the polarization properties of the biological samples. Taking the measurement results of the DRRP-CR at a strong light field as the standard, and the ME of Mueller matrix elements of the two biological tissues measured by the SMMP-CR were 0.0629 and 0.0726, as shown in Table 12. The RMS of the Mueller matrix of two samples measured by the SMMP-CR were 0.0334 and 0.0290, respectively. The differences measured by the DRRP-CR and SMMP-CR may be attributed to the following reasons. First, owing to the weak intensity of backscattering irradiances from thick biological tissues, removing the influence of background noise on polarization measurements by only calibration is challenging. In addition, the mean values of the Mueller matrix calculated by both systems are slightly different due to the difference of data processing by DRRP-CR and SMMP-CR. The SMMP-CR only can calculate the numerical solution of Mueller matrix instead of analytical solution by solving the optimization problem in Eq. (15). The Mueller matrix measured by SMMP-CR constantly approaches the optimal solution by using of MGA algorithm, but it is difficult to maintain consistency with theoretical values. The DRRP-CR can accurately obtain the analytic solution of the Mueller matrix of the sample on the basis of the intensity variation relationship of the 16 intensity distributions obtained at strong light field. Hence, the measurement results of DRRP-CR at strong light field are regarded as “gold standard” to verify the performance of polarization measurement by SMMP-CR at strong and low light filed.

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Table 12. Absolute error of Mueller matrix measured by SMMP-CR and DRRP-CR at strong light field

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Furthermore, the performances of the SMMP-CR and DRRP-CR at low light fields were tested by attenuating the electric current of the LED, which was approximately 100 times lower than that of a strong light field. Considering the ultra-low readout noise of the qCMOS, the NPBS3 and reflector2 were used to degrade the light intensity entering the qCMOS, which was four times lower than the intensity detected by the sCMOS. Thus, the performances of the DRRP-CR and SMMP-CR at low light fields were fairly compared. Figures 21(a) and (b) show Mueller matrix images of the pork and chicken breast tissues measured by the DRRP-CR at low light fields, respectively.

 figure: Fig. 21.

Fig. 21. Mueller matrix images of the biological samples at low light field, (a): pork tissue; (b): chicken breast tissue.

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Mueller matrix images disturbed by background noise at low light fields hardly discriminate the structural characteristics of biological tissues. Furthermore, the Mueller images at low light fields were acquired using the DRRP-CR after the 16 raw images were denoised, as shown in Fig. 22. Evidently, the denoised Mueller matrix images deviated from the images measured in a strong light field, and the Mueller matrix elements were overall close to zero. Therefore, the measurement of the Mueller matrix by the DRRP-CR was inaccurate at low light field. Next, the SMMP-CR was used to capture the overall polarization property of the sample at a low light field. Figures 23 and 24(a1)–(d1) show the raw four intensity distributions acquired by measuring the pork and chicken breast tissues at low light field, and Figs. 23 and 24 (a2)–(d2) show the corresponding denoised intensity distributions.

 figure: Fig. 22.

Fig. 22. (a) and (b) are the denoised Mueller matrix images of the biological samples at low light field, (c) and (d) are the denoised total intensity images of the biological samples at low light field; (a) and (c): pork tissue; (b) and (d): chicken breast tissue.

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 figure: Fig. 23.

Fig. 23. Intensity distributions acquired with the pork tissue as tested sample at a low light field, (a1) -15°, (b1) -52°, (c1) 52°, (d1) 15°, and (a2)–(d2) are the final denoised images corresponding to (a1)–(d1).

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 figure: Fig. 24.

Fig. 24. Intensity distributions acquired with the chicken breast tissue as tested sample at a low light field, (a1) -15°, (b1) -52°, (c1) 52°, (d1) 15°, and (a2)–(d2) are the final denoised images corresponding to (a1)–(d1).

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In addition, the background noise, including the surface-reflected light without illuminating the sample, was acquired, as shown in Fig. 25. Evidently, compared with the four intensity distributions acquired in the strong light field, the shape of the denoised four intensity distributions was still approximately maintained, although the measured raw intensity distributions had a low LSNR. The denoised images were used to calculate the Mueller matrix of the samples.

 figure: Fig. 25.

Fig. 25. Background noise acquired at a low light field, (a) −15°, (b) −52°, (c) 52°, (d) 15°.

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The Mueller matrices of the two types of samples were measured using the DRRP-CR and SMMP-CR at low light field, as presented in Table 13 and Table 14. Taking the measurement results of the DRRP-CR at strong light field as the gold standard, and the ME of Mueller matrix of the two biological tissues measured by the DRRP-CR at low light field were 0.7546 and 0.8056, as shown in Table 15. The RMS of the Mueller matrix of two samples were 0.3148 and 0.3340, respectively. Evidently, the measurement of the Mueller matrix by the DRRP-CR was inaccurate at low light fields. Compared with the standard, the two types of biological tissues were measured using the SMMP-CR at a low light field, and the ME of the Mueller matrix were 0.0856 and 0.0934, as shown in Table 16. The RMS of the Mueller matrix of two samples were 0.0433 and 0.0481, respectively. Hence, the SMMP-CR can significantly improve the accuracy of polarization measurements compared with the DRRP-CR at low light field.

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Table 13. The mean value of Mueller matrix of samples measured by DRRP-CR at low light field

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Table 14. The mean value of Mueller matrix of samples measured by SMMP-CR at low light field

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Table 15. Absolute error of Mueller matrix measured by DRRP-CR at low and strong light field

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Table 16. Absolute error of Mueller matrix measured by SMMP-CR at low light field and DRRP-CR at strong light field

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The local surface normal of sample will change over time in the process of acquisition by the DRRP-CR and SMMP-CR, which may change the polarization characteristics of biological tissues. Due to nonimaging characteristic of SMMP-CR, we can not directly contrast the change of Mueller matrix images measured by the DRRP-CR and SMMP-CR. To quantitatively identify the changes of polarization characteristics of biological tissues, we used DRRP-CR to measure chicken breast tissue twice, and obtained the Mueller matrix images and total intensity images (${m_{00}}$ of unnormalized Mueller matrix) from both measurements, as shown in Fig. 26. The two measurements of Mueller matrix by DRRP-CR required about 160s, and it takes more time than the total measurement processes of DRRP-CR and SMMP-CR. The maximal absolute error of Mueller matrix of two measurements is less than 0.02, as shown in Table 17.

 figure: Fig. 26.

Fig. 26. The normalized Mueller matrix images and total intensity images (${m_{00}}$) measured by DRRP-CR twice. (a) and (b) are the Mueller matrix images in two measurements; (c) and (d) are the total intensity images in two measurements;

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Table 17. The mean value and absolute error of Mueller matrix in two measurements

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Hence, we can see that polarization characteristics of biological tissues hardly changed in a short time, which may be attributed to the following reasons. First, biological tissue has been partially adsorbed by clean cloth and then it is placed on the sample table for a while to keep stable before measurement. Secondly, the experiment was conducted in ultra-clean room with constant humidity and temperature, and the local surface normal of sample little changed in a short time. In general, the total measurement processes of DRRP-CR and SMMP-CR required less than 100s, in which the polarization characteristics of biological tissues have little change.

The DRRP can indisputably achieve high-accuracy measurement of the Mueller matrix in a strong light field, which is regarded as the gold standard in the field of polarimetry. The SMMP has a shortcoming in that it is applicable only for acquiring the polarization properties of an isotropic sample or the overall polarization effect in a small area because of the nonimaging characteristic of SMMP. Unless the SMMP is improved by scanning the sample point by point, the DRRP is the best choice for capturing Mueller matrix images at strong light field with high resolution and high precision. The proposed SMMP was designed to satisfy the demand for measuring the polarization properties of samples with high speed and accuracy at low light fields, and it may be an important tool for the following requirements. First, when polarization measurement is implemented in a field with weak echo signals, such as remote sensing, quantum optics, and biomedical diagnosis. Second, when polarization measurement with weak illumination is required because irradiation by strong light damages or changes the polarization properties of the sample, for example, in the measurement of living biological tissues and micro-nano materials. In this study, the Mueller matrix of thick biological tissues was obtained to verify the reliability of the SMMP-CR for assessing the polarization properties of a sample whose Mueller matrix was unknown beforehand. The SMMP-CR was designed to capture the polarization properties from the backscattered light of thick biological tissues; further, it was compared with the DRRP-CR and found to exhibit excellent performance at low light fields.

5. Conclusion

In this study, inspired by spatial polarization modulation, a ZVQR was placed in front of the PSA to constitute an SMPSA that can map the polarization state of the outgoing light of the sample to an intensity distribution. Such simplified Mueller polarimeters can achieve fast acquisition of the Mueller matrix of a sample using only four camera shots rather than 16 shots, as in the state of the art technique. In addition, an AHTF combined with an LFFT filter was developed by considering the difference in LSNR for the different intensity distributions to denoise the raw intensity distributions at low light fields. The proposed method could accurately reconstruct the Mueller matrix by analyzing the distribution characteristics of photons in the final denoised images. Moreover, the MGA algorithm was proposed to expedite convergence efficiency in the reconstruction of the Mueller matrix by iterative optimization. The RMS and ME of Mueller matrix of sample measured by the SMMP at low light field were smaller than 0.05 and 0.09 as shown in Table 6, which indicates that the SMMP can still measure the Mueller matrix with high accuracy at low light fields. While the RMS and ME of Mueller matrix of sample measured by the DRRP were approximate to 0.3 and 0.8 as shown in Table 7 and 8. The experimental results revealed that the proposed SMMP could significantly improve the measurement accuracy of the Mueller matrix over the DRRP at a low light field by a factor of approximately 10. In addition, the SMMP-CR was designed to capture the polarization properties of a sample whose Mueller matrix was unknown beforehand; in this regard, it demonstrated excellent performance compared with the DRRP-CP at low light field. Benefiting from the improvement in speed and accuracy for the measurement of the Mueller matrix at low light fields, the SMMP may provide a potential approach for the characterization of the physical properties in some applications, ranging from remote sensing to diagnosis of living biological tissues.

Funding

National Natural Science Foundation of China (62175014, 61975013); National Science and Technology Major Project (2017ZX02101006-001); Major Scientific Instrument Development Project of National Natural Science Foundation of China (11627808).

Acknowledgments

We gratefully acknowledge the National Science Foundation for their help in identifying collaborators for this work.

Disclosures

The authors declare that there are no conflicts of interest related to this article.

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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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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Figures (26)
Fig. 1.
Fig. 1. Flowchart for the proposed MGA algorithm, and $|{\kern 1pt} {\kern 1pt} \cdot {\kern 1pt} {\kern 1pt} |$ is the absolute value function, and the number of iterations is represented as ${k_{\max }}$.

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Fig. 2.
Fig. 2. The convergence curves of different optimization algorithm.

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Fig. 3.
Fig. 3. Schematic of the experimental system of the spatially modulated Mueller polarimeter (SMMP). It included a collimated LED, polarization state generator (PSG), spatially modulated polarization state analyzer (SMPSA), linear horizontal polarizers P1 and P2, quarter-wave plates Q1 and Q2, zero-order vortex quarter wave retarder (ZVQR), focusing lens (Lens3), and scientific complementary metal-oxide-semiconductor (sCMOS).

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Fig. 4.
Fig. 4. The calibrated model parameters, (a): ${m_{00}}$, (b): ${m_{01}}$, (c): ${m_{02}}$, (d): ${m_{03}}$. The calculated pixel area is 80 × 80, and a circular mask is added to select the effective pixel area.

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Fig. 5.
Fig. 5. The measured four intensity distributions with air as tested sample, (a): -15°, (b): -52°, (c): 52°, (d):15°.

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Fig. 6.
Fig. 6. The theoretical four intensity distributions with air as sample, (a): -15°, (b): -52°, (c): 52°, (d):15°.

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Fig. 7.
Fig. 7. The measured four intensity distributions with QWP (0°) as tested sample, (a): -15°, (b): -52°, (c): 52°, (d):15°.

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Fig. 8.
Fig. 8. The theoretical four intensity distributions with QWP (0°) as sample, (a): -15°, (b): -52°, (c): 52°, (d):15°.

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Fig. 9.
Fig. 9. The Mueller matrix images of four sample measured by DRRP. (a): air, (b): QWP (0°), (c): QWP (45°), (d): QWP (90°).

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Fig. 10.
Fig. 10. Flow chart of the denoising process.

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Fig. 11.
Fig. 11. The LSNR of four intensity distributions of different samples.

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Fig. 12.
Fig. 12. The measured four raw-intensity distributions with air as tested sample at low light field, (a1): -15°, (b1): -52°, (c1): 52°, (d1): 15°, and (a2)- (d2) are the final denoised images corresponding to (a1) - (d1).

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Fig. 13.
Fig. 13. The measured four raw-intensity distributions with QWP (0°) as tested sample at low light field, (a1): -15°, (b1): -52°, (c1): 52°, (d1): 15°, and (a2)- (d2) are the final denoised images corresponding to (a1) - (d1).

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Fig. 14.
Fig. 14. (a) 16 raw-intensity distributions detected by the sCMOS; (b): intensity distribution of background noise; (c) LSNR of different intensity distributions ${I_1},{I_2}, \cdots {I_{16}}$.

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Fig. 15.
Fig. 15. Denoised Mueller images of four samples measured by the DRRP at low light field. (a): air, (b): QWP (0°), (c): QWP (45°), (d): QWP (90°).

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Fig. 16.
Fig. 16. Photographs (a) and schematic diagram (b) of the instrument. The same devices in the two figures are outlined by dotted lines of the same color.

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Fig. 17.
Fig. 17. Photographs of the biological samples: (a) pork tissue, (b) chicken breast tissue.

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Fig. 18.
Fig. 18. (a) and (b) are normalized Mueller matrix images of the biological samples; (c) and (d) are the total intensity images of the biological samples; (a) and (c): pork tissue; (b) and (d): chicken breast tissue.

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Fig. 19.
Fig. 19. Four intensity distributions acquired with pork tissue as the tested sample, (a) −15°, (b) −52°, (c) 52°, (d) 15°.

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Fig. 20.
Fig. 20. Four intensity distributions acquired with chicken breast tissue as the tested sample, (a) −15°, (b) −52°, (c) 52°, (d) 15°.

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Fig. 21.
Fig. 21. Mueller matrix images of the biological samples at low light field, (a): pork tissue; (b): chicken breast tissue.

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Fig. 22.
Fig. 22. (a) and (b) are the denoised Mueller matrix images of the biological samples at low light field, (c) and (d) are the denoised total intensity images of the biological samples at low light field; (a) and (c): pork tissue; (b) and (d): chicken breast tissue.

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Fig. 23.
Fig. 23. Intensity distributions acquired with the pork tissue as tested sample at a low light field, (a1) -15°, (b1) -52°, (c1) 52°, (d1) 15°, and (a2)–(d2) are the final denoised images corresponding to (a1)–(d1).

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Fig. 24.
Fig. 24. Intensity distributions acquired with the chicken breast tissue as tested sample at a low light field, (a1) -15°, (b1) -52°, (c1) 52°, (d1) 15°, and (a2)–(d2) are the final denoised images corresponding to (a1)–(d1).

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Fig. 25.
Fig. 25. Background noise acquired at a low light field, (a) −15°, (b) −52°, (c) 52°, (d) 15°.

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Fig. 26.
Fig. 26. The normalized Mueller matrix images and total intensity images (${m_{00}}$) measured by DRRP-CR twice. (a) and (b) are the Mueller matrix images in two measurements; (c) and (d) are the total intensity images in two measurements;

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Tables (17)

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Table 1. Mean values and absolute error of the Mueller matrix of samples measured by SMMP at strong light field

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Table 2. RMS and ME of the Mueller matrix of samples measured by SMMP at strong light field

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Table 3. Mean values and absolute error of the Mueller matrix of samples measured by DRRP at strong light field

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Table 4. RMS and ME of the Mueller matrix of samples measured by DRRP at strong light field

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Table 5. Mean values and absolute error of the Mueller matrix of samples measured by SMMP at low light field

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Table 6. RMS and ME of the Mueller matrix of samples measured by SMMP at low light field

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Table 7. Mean values and absolute error of the Mueller matrix of samples measured by DRRP at low light field

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Table 8. RMS and ME of the Mueller matrix of samples measured by DRRP at low light field

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Table 9. The measured mean values of Mueller matrix of mirror after calibration

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Table 10. The mean value of Mueller matrix of samples measured by DRRP-CR at strong light field

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Table 11. The mean value of Mueller matrix of samples measured by SMMP-CR at strong light field

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Table 12. Absolute error of Mueller matrix measured by SMMP-CR and DRRP-CR at strong light field

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Table 13. The mean value of Mueller matrix of samples measured by DRRP-CR at low light field

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Table 14. The mean value of Mueller matrix of samples measured by SMMP-CR at low light field

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Table 15. Absolute error of Mueller matrix measured by DRRP-CR at low and strong light field

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Table 16. Absolute error of Mueller matrix measured by SMMP-CR at low light field and DRRP-CR at strong light field

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Table 17. The mean value and absolute error of Mueller matrix in two measurements

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Equations (35)

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$${\mathop s\limits^ \wedge}_{out} = M \cdot {\mathop s\limits^ \wedge}_{in},$$
$${\mathop {\textrm{s}}\limits^ \wedge} _{\textrm{out}} = \left[ {\begin{array}{{@{}cccc@{}}} \overrightarrow {\textrm{s}}_{\textrm{out}}^1&\overrightarrow {\textrm{s}}_{\textrm{out}}^2&\overrightarrow {\textrm{s}}_{\textrm{out}}^3&\overrightarrow {\textrm{s}}_{\textrm{out}}^4 \end{array}} \right] \cdot \left[ {\begin{array}{{@{}cccc@{}}} {\textrm{s}_0^1}&0&0&0\\ 0&{\textrm{s}_0^2}&0&0\\ 0&0&{\textrm{s}_0^3}&0\\ 0&0&0&{\textrm{s}_0^4} \end{array}} \right],$$
$$[\begin{array}{{@{}cccc@{}}} {{I_1}}&{{I_2}}&{{I_3}}&{{I_4}} \end{array}] = [1{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} 0{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} 0{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} 0] \cdot {M_{SMPSA}} \cdot {\mathop s\limits^ \wedge} _{\textrm{out}},$$
$${M_{SMPSA}}\textrm{ = }\left[ {\begin{array}{{@{}cccc@{}}} {{m_{00}}}&{{m_{01}}}&{{m_{02}}}&{{m_{03}}}\\ {{m_{10}}}&{{m_{11}}}&{{m_{12}}}&{{m_{13}}}\\ {{m_{20}}}&{{m_{21}}}&{{m_{22}}}&{{m_{23}}}\\ {{m_{30}}}&{{m_{31}}}&{{m_{32}}}&{{m_{33}}} \end{array}} \right].$$
$$[\begin{array}{{@{}cccc@{}}} {{I_1}}&{{I_2}}&{{I_3}}&{{I_4}} \end{array}] = [\begin{array}{{@{}cccc@{}}} {{m_{00}}}&{{m_{01}}}&{{m_{02}}}&{{m_{03}}} \end{array}] \cdot \left[ {\begin{array}{{@{}cccc@{}}} {\overrightarrow s_{out}^1}&{\overrightarrow s_{out}^2}&{\overrightarrow s_{out}^3}&{\overrightarrow s_{out}^4} \end{array}} \right] \cdot \left[ {\begin{array}{{@{}cccc@{}}} {s_0^1}&0&0&0\\ 0&{s_0^2}&0&0\\ 0&0&{s_0^3}&0\\ 0&0&0&{s_0^4} \end{array}} \right].$$
$$s_0^i = \sum {{I_i}} /\sum {([\begin{array}{{@{}cccc@{}}} {{m_{00}}}&{{m_{01}}}&{{m_{02}}}&{{m_{03}}} \end{array}] \cdot \overrightarrow s _{out}^i)} ,$$
$$[I_\textrm{1}^{air}{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} I_2^{air}{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} I_3^{air}{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} I_4^{air}] = [\begin{array}{{@{}cccc@{}}} {{m_{00}}}&{{m_{01}}}&{{m_{02}}}&{{m_{03}}} \end{array}] \cdot {M_{air}} \cdot [{\overrightarrow s _1}{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\overrightarrow s _2}{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\overrightarrow s _3}{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\overrightarrow s _4}],$$
$$[\begin{array}{{@{}cccc@{}}} {{m_{00}}}&{{m_{01}}}&{{m_{02}}}&{{m_{03}}} \end{array}] = [I_\textrm{1}^{air}{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} I_2^{air}{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} I_3^{air}{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} I_4^{air}] \cdot {[{\overrightarrow s _1}{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\overrightarrow s _2}{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\overrightarrow s _3}{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\overrightarrow s _4}]^{ - 1}}.$$
$${O_\textrm{i}} = \frac{{{O_\textrm{i}}}}{{\sum\limits_{\textrm{i = 1}}^N {{O_\textrm{i}}} }}{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} ,{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {I_\textrm{i}} = \frac{{{I_\textrm{i}}}}{{\sum\limits_{\textrm{i = 1}}^N {{I_\textrm{i}}} }},$$
$$NSSIM(O,I) = {(l(O,I))^\alpha } \cdot {(c(O,I))^\beta } \cdot {(s(O,I))^\gamma },$$
$$l(O,I) = \frac{{2{\mu _O}{\mu _I}}}{{\mu _O^2 + \mu _I^2}},$$
$$c(O,I) = \frac{{2{\sigma _O}{\sigma _I}}}{{\sigma _O^2 + \sigma _I^2}},$$
$$s(O,I) = \frac{{2{\sigma _{OI}}}}{{{\sigma _O}{\sigma _I}}},$$
$$NSSIM(O,I) = \frac{{6{\mu _O}{\mu _I}{\sigma _{OI}}}}{{(\mu _O^2 + \mu _I^2)(\sigma _O^2 + \sigma _I^2)}}.$$
$$\begin{aligned} {\overrightarrow s _{\max }} &= \mathop {\arg \max }\limits_{{{\overrightarrow s }_{out}}} (NSSIM(O,I))\\ &= \mathop {\arg \max }\limits_{{{\overrightarrow s }_{out}}} F(\overrightarrow s ). \end{aligned}$$
$$F_{{s_1}}^k = F(s_1^k,s_2^{k - 1},s_3^{k - 1}),$$
$$F_{{s_2}}^k = F(s_1^{k - 1},s_2^k,s_3^{k - 1}),$$
$$F_{{s_3}}^k = F_{{s_1}}^k(s_1^{k - 1},s_2^{k - 1},s_3^k),$$
$$s_i^k = s_i^{k - 1} + d_{{s_i}}^k,$$
$$d_{{s_i}}^k = g_{{s_i}}^k \cdot N_{{s_i}}^k + d_{{s_i}}^{k - 1} \cdot M_{{s_i}}^k,$$
$$g_{{s_i}}^k = \frac{{F_{{s_i}}^k - F_{{s_i}}^{k - 1}}}{{s_i^k - s_i^{k - 1}}}.$$
$$N_{{s_i}}^k = {{t_{{s_i}}^k \cdot sign(g_{{s_i}}^k)} / {g_{{s_i}}^k}},$$
$$M_{{s_i}}^k = {{r_{{s_i}}^k \cdot sign(d_{{s_i}}^{k - 1})} / {d_{{s_i}}^{k - 1}}},$$
$$M = {\mathop s\limits^ \wedge} _{in} \cdot inv({\mathop s\limits^ \wedge} _{in}),$$
$$\xi \textrm{ = }\frac{\textrm{m}}{2}\varphi \textrm{ + }\delta ,$$
$$LSNR\textrm{ = }10\cdot \log _{10}^{({E_r} - {E_B})/{\sigma _B}},$$
$$F(u,v) = \sum\limits_{x = 1}^M {\sum\limits_{y = 1}^N {O(x,y)} } {e^{ - i2\pi (ux/M + vy/N)}},$$
$$D(u,v) = \sqrt {{{(u - M/2)}^2} + {{(v - N/2)}^2}} ,$$
$$H(u,v) = \left\{ {\begin{array}{{c}} {1,{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} D(u,v) \le {d_0}}\\ {0,{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} D(u,v) > {d_0}} \end{array}} \right.{\kern 1pt} ,$$
$$F(u,v) = F(u,v) \cdot H(u,v).$$
$$O(x,y) = \frac{1}{{MN}}\sum\limits_{x = 1}^M {\sum\limits_{y = 1}^N {F(u,v)} } {e^{i2\pi (ux/M + vy/N)}}.$$
$$O(x,y) = O(x,y) - \frac{1}{{MN}}\sum\limits_{x = 1}^M {\sum\limits_{y = 1}^N {O(x,y)} } ,$$
$$O(x,y) = \left\{ {\begin{array}{{c}} {0,{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} O(x,y) < \gamma (LSNR)}\\ {{\kern 1pt} O(x,y) - \gamma (LSNR){\kern 1pt} {\kern 1pt} ,{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} O(x,y) \ge \gamma (LSNR){\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} } \end{array}} \right.{\kern 1pt} ,$$
$$\mathop \gamma \limits^ \wedge{=} \mathop {\arg \min }\limits_\gamma {\kern 1pt} {\kern 1pt} ({\overrightarrow s _M} - {\overrightarrow s _T}),$$
$$[\begin{array}{{@{}cccc@{}}} {{m_{00}}}&{{m_{01}}}&{{m_{02}}}&{{m_{03}}} \end{array}] = [I_1^{mirror} - {r_1}{\kern 1pt} I_2^{mirror} - {r_2}{\kern 1pt} I_3^{mirror} - {r_3} {\kern 1pt} I_4^{mirror} - {r_4}] \cdot {[{\overrightarrow s _1}{\kern 1pt} {\overrightarrow s _2}{\kern 1pt} {\overrightarrow s _3}{\kern 1pt} {\overrightarrow s _4}]^{ - 1}} \cdot M_{mirror}^{ - 1},$$
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