Polarization measurement accuracy analysis and improvement methods for the directional polarimetric camera

Chan Huang; Chan Huang; Chan Huang; Yuyang Chang; Yuyang Chang; Yuyang Chang; Guangfeng Xiang; Guangfeng Xiang; Guangfeng Xiang; Lin Han; Lin Han; Feinan Chen; Feinan Chen; Donggen Luo; Donggen Luo; Shuang Li; Shuang Li; Liang Sun; Liang Sun; Bihai Tu; Bihai Tu; Binghuan Meng; Binghuan Meng; Jin Hong; Jin Hong

doi:10.1364/OE.405834

1. Introduction

Aerosols [1–6] are important components in the atmosphere. They change the distribution of solar radiation in the atmosphere through the interaction with it, which has a great impact on the global radiation budget, atmospheric visibility and climate change [7–8]. In addition, aerosols are harmful to the health of human beings and other organisms by affecting air quality. Therefore, it is of great significance to study the characteristics of aerosols for improving climate conditions, controlling environmental pollution and promoting human health. However, as one of the most complex components in the atmosphere [9], aerosol particles are highly uncertain and need to be described by a large number of parameters such as particle sizes, morphologies, concentrations, absorption scattering characteristics and spatial distribution. A large number of theoretical studies have concluded that compared with traditional remote sensing methods, polarization remote sensing technique can provide more aspects of aerosol features to meet the needs of many important applications [10–11]. The directional polarimetric camera (DPC) [12–15], developed by Anhui Institute of Optics and Fine Mechanics (AIOFM), Chinese Academy of Sciences (CAS), is a space-borne polarimetric sensor with the POLDER (Polarization and Directionality of the Earth's Reflectances)-like design, which has the characteristics of ultra-wide angle and low-distortion imaging. It acquires the two-dimensional image of the earth with large field of view (${\pm} \textrm{50}$ degrees along-track, ${\pm} \textrm{50}$ degrees cross-track) and high spatial resolution (3.3 km) in 8 spectral bands from visible to near infrared, 3 of which are polarized spectral bands. The DPC could simultaneously conduct spectral, angular, and polarimetric measurements of atmospheric radiation, to theoretically maximize the sensitivity of observation results to aerosol characteristics [16–19]. The first DPC has been successfully launched on May 9, 2018 aboard the GaoFen-5 (GF-5) satellite, subject to Chinese high-resolution earth observation program. Four more DPCs will be installed on GF-5 (02), CM, DQ-1 and DQ-2 satellites to be launched successively from 2021 to 2022, respectively. Among them, the earliest satellite to be launched is GF-5 (02), a sun-synchronous orbiting satellite, at an altitude of 705km with an inclination of $\textrm{9}{\textrm{8}{^\circ} }$, which has 1:30 p.m. local overpass time and a two-day revisiting period. It is planned to be launched in 2021, with a mission duration of 8 years. The goal is to realize hyperspectral detection of the atmosphere and land. Although the first DPC and the DPC to be boarding the GF-5 (02) satellite have a strong similarity, compared with the first DPC, the DPC to be boarding the GF-5 (02) satellite has a series of improvements. For example, its focal length and effective pixels of the CCD are increased from 4.833mm to 5.540mm and from $512 \times 512$ to $1024 \times 1024$ respectively to provide higher spatial resolution. Its lenes employ the high refractive index material to make the incident angle of the light relative to the lens is less than 35 degrees, and its coating is optimized to reduce the diattenuation of the optics.

Polarization measurement accuracy [20–25] indicates the accuracy of the instrument in measuring the polarization state of the incident light. It is an important index to evaluate the performance of the DPC, which has the most direct impact on the reliability of some aerosol parameters inversion, such as complex refractive index, particle size and shape, since polarization is more sensitive to these parameters than radiation is [26–27]. Generally speaking, the polarization measurement accuracy of instruments is mainly related to the random error (i.e. the signal-to-noise ratio (SNR) of the measured image) [23–25] and the systematic error (i.e. the manufacturing error and the adjustment error of the components). To improve the polarization measurement accuracy of the instrument, in addition to using a series of preprocessing algorithms to improve the SNR of the measured image, it is also essential to perform high-precision polarimetric calibration on the instrument. Although the POLDER [3–4,28–32], multi-viewing-channel-polarization imager (3MI) [33–35], and hyper-angular rainbowp (HARP) [36] have the same type of large field of view polarimetric imager with DPC, there are few researches on polarimetric calibration based on clear physical definitions for this type of instrument. Bret-Dibat et. al. [28] first proposed an instrumental radiometric model considering the polarization effect for POLDER, and listed the calibration methods and results of each parameter without specific details. Based on those, Huang et. al. [12] derived the radiometric model of DPC in non-polarized bands and polarized bands in detail by using the Stokes vector and Mueller matrix, and described the specific calibration methods, equipment, and main results with related accuracies of each parameter. The results of verification experiments proved that the polarization detection accuracy of the DPC meets the requirements of data inversion for polarimetric sensor.

However, when calibrating the DPC to be boarding the GF-5 (02) satellite recently, we found that the on-ground calibration processes for some parameters in the previous calibration work [12] for the DPC onboard the GF-5 satellite was so idealized that some factors causing systematic errors are ignored. When the degree of linear polarization (DoLP) of incident light is calculated, the parameters with certain errors are used, which will affect the polarization measurement accuracy of the instrument. More specifically, the relative transmission of the polarized channels used to be obtained by calculating the ratio of the response of the three channels in the region of central field of view, and the diattenuation of the optics corresponding to full field of view angles used to be obtained by performing a polynomial fitting on the measured diattenuation corresponding to the different field of view angles. These calibration processes are based on the assumptions that the nonuniformity of the filter can be ignored and the diattenuation of the optics is circularly symmetric, respectively. However, limited by the manufacturing and processing level of components, the actual state of the DPC may deviate from the assumed state enough to affect the polarization measurement accuracy. In addition, the absolute azimuth angle accuracy of the second polarized channel is ${\pm} 1$ degree due to the influence of the adjustment error and machining accuracy of Glan-Taylor prism and straight-edge structure, while the relative azimuth angle accuracy of three polarized channels is better than ${\pm} 0.1$ degree since the least square method is used. The calibration error of the relative azimuth angle of the three polarized channels will cause the error in solving DoLP, which has a certain impact on the polarization measurement accuracy.

In this paper, after the analysis of the relationship between the polarization measurement accuracy of DPC and the parameter calibration errors caused by the nonideality of the components of DPC, a series of simple and practical methods are proposed to improve the calibration accuracy of the parameters. The methods include increasing the sampling points and obtaining the diattenuation of the optics corresponding to each pixel by interpolation, establishing the objective equation and using the optimization algorithm [37–39] to obtain the optimal solution of absolute azimuth angles, and calculating the relative transmission corresponding to each pixel (a matrix) to replace the original relative transmission corresponding to central field of view (a number), which enable DPC to achieve higher polarization measurement accuracy. The rest of the paper is organized as follows. Firstly, the instrumental concept and radiometric model of DPC is presented in Section 2. Secondly, the main factors that cause the polarization measurement error of DPC are analyzed in Section 3. Then, methods and equipment for improving polarization measurement accuracy of DPC are described in detail in Section 4. Finally, polarization measurement accuracy verification experiments were carried out to verify the effectiveness of these methods. The results and discussions are shown in Section 5. Conclusions are drawn in Section 6.

2. DPC instrumental concept and radiometric model

2.1 DPC instrumental concept

A brief presentation of the DPC instrumental concept and the radiometric model is necessary to understand the instrument. The more detailed description of the instrument has been presented in [12]. The DPC instrumental concept is based on a large field of view telecentric optics, a rotating wheel module carrying spectral filters and polarizers, and a two-dimensional charge coupled device (CCD) matrix array detector. Figure 1 shows the optics on a meridional plane of DPC.

Fig. 1. Optics of the DPC.

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The large field of view [40–42] telecentric optics consists of an inverse Galileo telescope with large-aperture negative lenses followed by a focusing group which is composed of three double-cemented positive lenses. The main function of the inverse Galileo telescope is to reduce the angle between the beam and the optical axis and to eliminate astigmatism and distortion of the system as much as possible. The focusing group is used to balance the residual astigmatism, distortion and other aberrations of the inverse Galileo telescope and focus the beam. A diaphragm aperture is placed between the telescope and the focusing group, so that the lens is telecentric in the image space. The large field of view optics design of DPC makes it possible to provides as much as 9 viewing directions and 126, i.e. ($9 \times (5 + 3 \times 3)$) observation vectors for an observed object, as explained in the following.

The multi-polarization and multi-spectral capability of DPC is provided by the rotating wheel module, which includes five non-polarized bands (443, 565, 763, 765 and 910nm) and three polarized bands (490, 670 and 865nm). Since only linear polarization detection is involved, three continuous channels are set for each polarized band on the rotating wheel. Each channel corresponding to one polarized band is composed of the identical spectral filter and linear polarizer which is fixed at different relative azimuth angles (0 degree, 60 degrees, 120 degrees) for different channels. To compensate for the DPC motion during the delay due to the three polarized measurements and to co-register the polarized images, a wedge prism is added to the first and third channel, respectively. For the wedge prism, POLDER and subsequent 3MI have dropped this concept mainly because the non-ideality of the wedge prism will affect the geometrical registration accuracy between the three polarized channels. However, it will cause the same target imaged on different pixels of the three polarized channels of the instrument. Thus, it is necessary to correct the response inconsistency of the CCD frequently to avoid the polarization measurement error caused by it. With this in mind, the wedge prism has not been dropped by DPC. According to the characteristics of DPC, the whole imaging optical system meets the circularly symmetric design.

2.2 DPC radiometric model

The objective of establishing the radiometric model [12,28,43–45] of the DPC is to give a complete and totally representative description of the physical properties of the instrument, thereby characterizing the response of each pixel of the CCD detector matrix in each spectral channel to any input polarized light. The instrument reference frame O₀-X₀Y₀ with the center of the CCD as the origin and local reference frame O-XY with the direction of parallel tangential plane as the X axis, the direction of vertical tangential plane as the Y axis, and the certain pixel as the origin are established, as shown in Fig. 2. The coordinates of the certain pixel could be expressed as $\textrm{(}i,\textrm{ }j\textrm{)}$ or $(\theta ,\textrm{ }\phi )$ and obtained by geometric calibration. $\theta $ is the field of view angle and $\phi $ is the azimuth angle. The angle between the polarization vector E and the X axis is the angle of linear polarization (AoLP) $\chi $, while the angle between the polarizer transmission axis and the X₀ axis is absolute azimuth angle $\alpha $.

Fig. 2. Reference frames and the polarizer direction; (X0, Y0) for the instrument reference frame; (X, Y) for the local reference frame.

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The derivation process of the radiometric model with polarization parameters of the DPC has been presented in an early companion paper [12]. For linearly polarized incident light defined by its Stokes parameters $\textrm{(I, Q, U)}$, in the polarized spectral band of DPC, the output signal for the pixel with the coordinates of $\textrm{(}i,\textrm{ }j\textrm{)}$ is:

(1)$$DN_{i,j}^{k,a} = G \cdot t \cdot {A^k} \cdot {T^{k,a}} \cdot P_{i,j}^k \cdot ({P1_{i,j}^{k,a}I_{i,j}^k + P2_{i,j}^{k,a}Q_{i,j}^k + P3_{i,j}^{k,a}U_{i,j}^k} )+ C_{i,j}^k$$

where G is the relative gain coefficient, t the exposure time, whose values reference the nominal gain (${\textrm{G}_{\textrm{nominal}}}\textrm{ = }1$) and original exposure time (${\textrm{t}_{\textrm{original}}}\textrm{ = 1}$). ${\textrm{A}^\textrm{k}}$ is the absolute calibration coefficient of the spectral band $\textrm{k}$, which is expressed in $\textrm{DN} \cdot \mathrm{\mu}{\textrm{W}^{ - 1}} \cdot \textrm{c}{\textrm{m}^\textrm{2}} \cdot \textrm{sr} \cdot \textrm{nm}$ and determined by the reference pixel at the center of the CCD matrix. ${\textrm{T}^{\textrm{k,a}}}$ is the relative transmission of the polarized channel a affiliated to the spectral band $\textrm{k}$, the second polarized channel can be used as the reference and its value is set to 1. $\textrm{P}_{\textrm{i,j}}^\textrm{k}$, the total relative response variation, indicates the inconsistency of the response of the different fields of view of the DPC to the same incident radiation. $\textrm{C}_{\textrm{i,j}}^\textrm{k}$ denotes the dark current. $\textrm{P1}$, $\textrm{P2}$ and $\textrm{P3}$ indicate the effect of lens, filter and polarizer on polarization, which can be described as

(2)$$\left\{ {\begin{array}{{l}} {P1_{i,j}^{k,a} = 1 + \varepsilon_{i,j}^k \cdot \textrm{cos}[2({\alpha^{k,a}} - \phi )]}\\ {P2_{i,j}^{k,a} = \varepsilon_{i,j}^k + \textrm{cos}[2({\alpha^{k,a}} - \phi )]}\\ {P3_{i,j}^{k,a} = \sqrt {1 - \varepsilon {{_{i,j}^k}^{^2}}} \cdot \textrm{sin}[2({\alpha^{k,a}} - \phi )]} \end{array}} \right.$$

where $\mathrm{\varepsilon} _{\textrm{i,j}}^\textrm{k}$ represents the diattenuation of the optics of the spectral band $\textrm{k}$. Although the wedge prism is added to the first and third channel of the polarized spectral band, its influence on the polarization state of incident light can be ignored compared with other components, and needn’t to be considered in the radiometric model. It has been explained and discussed in detail in the appendix.

For a non-polarized band, the difference of the above model is the relative transmission against the polarized channels and the effect of polarizers on polarization are not applicable. Therefore, the radiometric model of the non-polarized channels of the DPC can be described as

(3)$$DN_{i,j}^k = G \cdot t \cdot {A^k} \cdot P_{i,j}^k \cdot ({P1_{i,j}^kI_{i,j}^k + P2_{i,j}^kQ_{i,j}^k + P3_{i,j}^kU_{i,j}^k} )+ C_{i,j}^k$$

where $\textrm{P1}_{\textrm{i,j}}^\textrm{k}\textrm{ = 1}$, $\textrm{P2}_{\textrm{i,j}}^\textrm{k} = \mathrm{\mathrm{\varepsilon}} _{\textrm{i,j}}^\textrm{k}$ and $\textrm{P3}_{\textrm{i,j}}^\textrm{k}\textrm{ = 0}$.

3. DPC polarization measurement accuracy analysis

Before introducing the methods for improving the polarization measurement accuracy of the instrument, it is necessary to explore the main factors that cause the polarization measurement error of DPC and evaluate their magnitude. DPC could obtain the polarization information of the observed object by simultaneously solving the measurements of the three polarized channels in the same polarized spectral band in combination with the radiometric model. The relationship between the measurements and the Stokes parameters of the incident light is shown as

(4)$$\left[ {\begin{array}{{c}} {DC_{i,j}^{k,1}}\\ {DC_{i,j}^{k,2}}\\ {DC_{i,j}^{k,3}} \end{array}} \right] = M_{i,j}^k \cdot \left[ {\begin{array}{{c}} {I_{i,j}^k}\\ {Q_{i,j}^k}\\ {U_{i,j}^k} \end{array}} \right]$$

where $DC_{i,j}^{k,a}\textrm{ = }DN_{i,j}^{k,a} - C_{i,j}^k$, $M_{i,j}^k$ is the measurement matrix and can be described as

(5)$$M_{i,j}^k = G \cdot t \cdot {A^k} \cdot P_{i,j}^k \cdot \left[ {\begin{array}{{ccc}} {{T^{k,1}} \cdot P1_{i,j}^{k,1}}&{{T^{k,1}} \cdot P2_{i,j}^{k,1}}&{{T^{k,1}} \cdot P3_{i,j}^{k,1}}\\ {{T^{k,2}} \cdot P1_{i,j}^{k,2}}&{{T^{k,2}} \cdot P2_{i,j}^{k,2}}&{{T^{k,2}} \cdot P3_{i,j}^{k,2}}\\ {{T^{k,3}} \cdot P1_{i,j}^{k,3}}&{{T^{k,3}} \cdot P2_{i,j}^{k,3}}&{{T^{k,3}} \cdot P3_{i,j}^{k,3}} \end{array}} \right]$$

Without considering the SNR of the instrument, the polarization measurement accuracy of the instrument is mainly related to the accuracy of calibration parameters. The polarization information obtained by inversion can be expressed as

(6)$$\overline I _{i,j}^k \cdot \left[ {\begin{array}{{c}} 1\\ {\overline {DoLP} \cdot \cos ({2\overline \chi } )}\\ {\overline {DoLP} \cdot \sin ({2\overline \chi } )} \end{array}} \right] = I_{i,j}^k \cdot {(\overline M _{i,j}^k)^{ - 1}} \cdot \hat{M}_{i,j}^k \cdot \left[ {\begin{array}{{c}} 1\\ {DoLP \cdot \cos 2\chi }\\ {DoLP \cdot \sin 2\chi } \end{array}} \right]$$

where $\overline I _{i,j}^k$, $\overline {DoLP}$ and $\overline \chi $ represent the total radiation intensity, degree of linear polarization and AoLP of the observed object obtained by inversion, respectively, while $I_{i,j}^k$, $DoLP$ and $\chi$ represent the actual polarization information of the incident light. $\overline M _{i,j}^k$ is the measurement matrix obtained by calibration, while $\hat{M}_{i,j}^k$ is the actual measurement matrix, which represents the real state of the instrument. Through the analysis of Eq. (6), it can be seen that the main factor that causes the inversion error is the difference between $\overline M$ and $\hat{M}$. As the DoLP is a relative quantity, its calculation is only related to the parameters T, $P1$, $P2$ and $P3$ in M, that is, related to the calibration errors of the parameters $\mathrm{\varepsilon}$, T, and $\alpha$, and the magnitude of the inversion error is also related to the DoLP and the AoLP of the incident light. To explain this problem more specifically, taking the DPC to be carried on GF-5 (02) satellite as an example, the polarization measurement processes of the observed object at the pixel A in the central field of view and the pixel B in the edge field of view are simulated. The set values of the calibration parameters in measurement matrix $\hat{M}$ corresponding to the two pixels are shown in Table 1.

Table 1. Values of the calibration parameters of two pixels.

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By changing the calibration results of the parameters in measurement matrix $\overline M$ to represent the calibration error, the difference between the $\overline {DoLP}$ and $DoLP$ can be calculated to explore the influence of the calibration errors of these three parameters on the $\overline {DoLP}$. The assumed calibration errors are in accordance with the characteristics of the instrument.

3.1. Diattenuation of the optics

The diattenuation of the optics corresponding to the full field angle is generally obtained by polynomial fitting of the measured diattenuation corresponding to several different field angles, which is based on the assumption that the optical system of the DPC is axisymmetric and the diattenuation of the optics is circularly symmetric. However, limited by lens manufacturing and coating, the diattenuation cannot be completely circularly symmetric, which results in the calibration error. The measurement error of the DoLP caused by the calibration error of $\mathrm{\varepsilon}$ can be expressed as

(7)$$\delta DoL{P_{\varepsilon} } = {\left. {\frac{{\partial \overline {DoLP} }}{{\partial \overline {\varepsilon} }}} \right|_{\overline {\varepsilon} = {\varepsilon} + \delta {\varepsilon} ,DoLP,\chi }} \cdot \delta {\varepsilon}$$

where $\delta DoLP$ is the measurement error of DoLP, $\delta {\varepsilon} $ the calibration error of ${\varepsilon}$. Pixel A is located in the central field of view, and the set value of ${\varepsilon}$ is only 0.003. Therefore, the absolute calibration error of ${\varepsilon}$ must be small, assumed to be ${\pm} 0.001$. While pixel B is located in the edge field of view, the ${\varepsilon}$ is large and greatly affected by the system asymmetry, so the original calibration method will have a larger error. According to the calibration experiments and uncertainty analysis of the parameters (discussed in detail in the next section, the results are shown in Fig. 9), for pixel B whose ${\varepsilon}$ is 0.055, it is a suitable assumption to set the calibration error to ${\pm} 0.005$. The polarization measurement errors of the instrument to the incident light with different DoLPs and AoLPs caused by $\delta {\varepsilon} $ are calculated, and the results are shown in Figs. 3(a), 3(b), 3(c), and 3(d), respectively.

Fig. 3. Polarization measurement errors of the instrument corresponding to the incident light with different DoLPs and AoLPs caused by $\delta {\varepsilon}$ (a) In the central field of view, ${\varepsilon} \textrm{ = }0.003$ and $\delta {\varepsilon} \textrm{ = }0.001$. (b) In the central field of view, ${\varepsilon} \textrm{ = }0.003$ and $\delta {\varepsilon} \textrm{ ={-} }0.001$. (c) In the edge field of view, ${\varepsilon} \textrm{ = }0.055$ and $\delta {\varepsilon} \textrm{ = }0.005$. (d) In the edge field of view, ${\varepsilon} \textrm{ = }0.055$ and $\delta {\varepsilon} \textrm{ ={-} }0.005$.

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As can be seen from Fig. 3, with the change of DoLP and AoLP of the incident light, the polarization measurement errors of pixel A and pixel B also change, and their changing trends are the same. The maximum value of polarization measurement error is almost the same as the set value of $\delta {\varepsilon} $.

3.2. Relative transmission of the polarized channels

Due to the negligible polarization effect caused by the lenses in the central field of view, the relative transmission of the polarized channels was obtained by calculating the ratio of the response of the three channels in the region of central field of view to non-polarized light. The second polarized channel is usually used as the reference. However, because the nonuniformity of the filter is not considered, such a calibration method can only guarantee the accuracy of T in the central field of view rather than the edge field of view. For example, the nonuniformity of filters used in DPC is ${\pm} 0.5\%$, that is, the maximum transmission is 0.5% higher than the median, and the lowest transmission is 0.5% lower than the median. If for the first channel, the transmission is the lowest in the central field of view and the highest in the edge field of view, while for the second channel, it is the lowest in the edge field of view and the highest in the central field of view and moreover, they keep the same transmission in the central field of view, the worst case will occur, where the calibration error of T in the edge field of view can even reach approximately $2\%$. The measurement error of the DoLP caused by the calibration error of T can be expressed as

(8)$$\delta DoL{P_T} = {\left. {\frac{{\partial \overline {DoLP} }}{{\partial \overline {{T^1}} }}} \right|_{\overline {{T^1}} = {T^1} + \delta {T^1},DoLP,\chi }} \cdot \delta {T^1} + {\left. {\frac{{\partial \overline {DoLP} }}{{\partial \overline {{T^3}} }}} \right|_{\overline {{T^3}} = {T^3} + \delta {T^3},DoLP,\chi }} \cdot \delta {T^3}$$

where $\delta {T^a}$ is the calibration error of ${T^a}$. Since the original calibration method obtain ${T^a}$ by measuring the relative response of three polarization channels in the central field of view of DPC, which results in a small $\delta {T^a}$ near the central field of view, while the $\delta {T^a}$ in the edge field is greatly affected by the nonuniformity of the filter. According to the calibration experiments (discussed in detail in the next and the results are shown in Fig. 15), the difference between the calibration results corresponding to the edge field of view and central field of view is manifest, reaching 0.013. Considering that the uncertainty of the improved calibration method is better than 0.2% and $\delta {T^1}$ and $\delta {T^3}$ can have same or opposite signs, the $(\delta {T^1},\textrm{ }\delta {T^3})$ for pixel A is set as $(0.003,\textrm{ }0.003)$ and $(0.003,\textrm{ - }0.003)$ in turn, while the $(\delta {T^1},\textrm{ }\delta {T^3})$ for pixel B is set as $(0.015,\textrm{ }0.015)$ and $(0.015,\textrm{ - }0.015)$ in turn. The polarization measurement errors of the instrument to the incident light with different DoLPs and AoLPs caused by $\delta {T^a}$ are calculated, and the results are shown in Fig. 4(a), 4(b), 4(c), and 4(d), respectively.

Fig. 4. Polarization measurement errors of the instrument corresponding to the incident light with different DoLPs and AoLPs caused by $\delta {T^a}$ (a) In the central field of view, $\delta {T^1}\textrm{ = }0.003$ and $\delta {T^3}\textrm{ = }0.003$. (b) In the central field of view, $\delta {T^1}\textrm{ = }0.003$ and $\delta {T^3}\textrm{ ={-} }0.003$. (c) In the edge field of view, $\delta {T^1}\textrm{ = }0.015$ and $\delta {T^3}\textrm{ = }0.015$. (d) In the edge field of view, $\delta {T^1}\textrm{ = }0.015$ and $\delta {T^3}\textrm{ ={-} }0.015$.

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It can be seen from Fig. 4 that the polarization measurement error of the pixel A is much smaller than that of the pixel B. The maximum polarization measurement error of the pixel A and pixel B is 0.0035 and 0.017, respectively. This is mainly because the $\delta {T^a}$ in the central field of view is much smaller than that in the edge field of view, which is consistent with the actual situation. When $\delta {T^1}$ and $\delta {T^3}$ are one positive and one negative, the resulting polarization measurement error is greater than the case where they are both positive.

3.3. Absolute azimuth angle

To obtain the absolute azimuth angle of three polarized channels, firstly, the absolute azimuth angle of P2 channel can be obtained by the Glan-Taylor prism and straight-edge structure. Then, the relative azimuth of three polarized channels are obtained by using a rotating polarizer. Finally, the absolute azimuth angle of the other two polarized channels can be obtained by the relative azimuth transfer. Due to the influence of the adjustment error and machining accuracy of the Glan-Taylor prism and straight-edge structure, the absolute azimuth accuracy is ${\pm} 1$ degree, while the relative azimuth accuracy is better than ${\pm} 0.1$ degree since the least square method is used to fit the collected data at multiple polarizer angles. The calibration errors of the relative azimuth angle of three polarized channels are also important factors affecting the polarization measurement accuracy of the instrument and the measurement error can be described as

(9)$$\delta DoL{P_\alpha } = \sum\limits_{a = 1}^3 {{{\left. {\frac{{\partial \overline {DoLP} }}{{\partial \overline {{\alpha^a}} }}} \right|}_{\overline {{\alpha ^a}} = {\alpha ^a} + \delta {\alpha ^a},DoLP,\chi }} \cdot \delta {\alpha ^a}}$$

where $\delta {\alpha ^a}$ is the calibration error of ${\alpha ^a}$. Because ${\alpha ^1}$ and ${\alpha ^3}$ are obtained through the relative azimuth angles of three polarizers and ${\alpha ^2}$, $\delta {\alpha ^1}$ and $\delta {\alpha ^3}$ include the calibration error of not only the relative azimuth angle but also the absolute azimuth angle of the second polarized channel. The maximum error of the original calibration method for ${\alpha ^a}$ is considered, that is, the absolute azimuth error of the second polarized channel is 1 degree, and the relative azimuth error of the first and third polarized channel is ${\pm} 0.1$ degree. Since the polarizers used in DPC are metal wire grid polarizer, the absolute azimuth angle of three polarized channels is independent of the pixel position, $(\delta {\alpha ^1},\textrm{ }\delta {\alpha ^2},\textrm{ }\delta {\alpha ^3})$ are set to be $(1.1,\textrm{ }1,\textrm{ }1.1)$ and $(1.1,\textrm{ }1,\textrm{ }0.9)$ for pixel A and B in turn. The calculating results of the polarization measurement errors of the instrument to the incident light with different DoLPs and AoLPs caused by $\delta {\alpha ^a}$ are shown in Figs. 5(a), 5(b), 5(c), and 5(d), respectively.

Fig. 5. Polarization measurement errors of the instrument corresponding to the incident light with different DoLPs and AoLPs caused by $\delta {\alpha ^a}$ (a) In the central field of view, $\delta {\alpha ^1}\textrm{ = }1.1$, $\delta {\alpha ^2}\textrm{ = }1$ and $\delta {\alpha ^3}\textrm{ = }1.1$. (b) In the central field of view, $\delta {\alpha ^1}\textrm{ = }1.1$, $\delta {\alpha ^2}\textrm{ = }1$ and $\delta {\alpha ^3}\textrm{ = }0.9$. (c) In the edge field of view, $\delta {\alpha ^1}\textrm{ = }1.1$, $\delta {\alpha ^2}\textrm{ = }1$ and $\delta {\alpha ^3}\textrm{ = }1.1$. (d) In the edge field of view, $\delta {\alpha ^1}\textrm{ = }1.1$, $\delta {\alpha ^2}\textrm{ = }1$ and $\delta {\alpha ^3}\textrm{ = }0.9$.

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As can be seen from Fig. 5, the polarization measurement error in the central field of view is slight smaller than that in the edge field of view. In these two settings, specifically, the maximum polarization measurement errors are 0.002 and 0.004 for the central field of view, and 0.0035 and 0.004 for the edge field of view, respectively. Similar to the effect of $\delta {T^a}$, the polarization measurement error corresponding to one positive and one negative $\delta {\alpha ^a}$ is larger than that corresponding to both positive $\delta {\alpha ^a}$.

In summary, the calibration error of T has the greatest influence on the polarization measurement accuracy of the instrument among the three parameters, and the polarization measurement errors in the central field of view caused by the calibration errors of all three parameters are smaller than that in the edge field of view, which is mainly due to the smaller calibration errors in the center field of view and the smaller ${\varepsilon}$. Because the three kinds of polarization measurement errors are independent of each other, the DPC polarization measurement errors caused by parameter calibration errors can be expressed as Eq. (10).

(10)$$\delta D\textrm{oLP} = \sqrt {{{({\delta D\textrm{oL}{\textrm{P}_{\varepsilon} }} )}^2} + {{({\delta D\textrm{oL}{\textrm{P}_\alpha }} )}^2} + {{({\delta D\textrm{oL}{\textrm{P}_T}} )}^2}}$$

The polarization measurement errors in the central field of view and edge field of view of DPC caused by parameter calibration errors are shown in Fig. 6. It can be seen that the maximum polarization measurement error is about 0.004 for the central field of view and 0.018 for the edge field of view, respectively. Of course, this does not mean that the polarization measurement error in the edge field of view of DPC will definitely reach 0.018, because in the simulation of parameter calibration errors, especially in the simulation of ${{\delta} T}^a$, the most extreme cases were considered. Considering that other factors such as the polarization characteristics of CCD, noise and stray light will also affect the polarization measurement accuracy of DPC, it is necessary to improve the calibration accuracy of these three main parameters to improve the polarization measurement accuracy as much as possible.

Fig. 6. Polarization measurement errors of the instrument corresponding to the incident light with different DoLPs and AoLPs caused by parameter calibration errors. (a) In the central field of view. (b) In the edge field of view.

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4. DPC polarization measurement accuracy improvement methods

As briefly mentioned in the introduction, through a series of simple methods, more accurate calibration results of the three main parameters can be obtained to improve the polarization measurement accuracy of DPC, which will be described in detail in this section.

4.1. Diattenuation of the optics

Among the three parameters, ${\varepsilon}$ needs to be calibrated firstly, because its calibration process does not require the other two parameters, while the calibration processes of the other two parameters need the accurate value of ${\varepsilon}$. The ${\varepsilon}$ of a certain point could be obtained by fitting the responses to the incident linear polarized light with different AoLPs with Eq. (11).

(11)$$DN_{i,j}^k = Z_{i,j}^k \cdot ({1 + {\varepsilon}_{i,j}^k \cdot \cos ({{{\pi \cdot ({\chi - {\chi_0}} )} / {90}}} )} )+ C_{i,j}^k$$

where $\textrm{Z = G} \cdot \textrm{t} \cdot \textrm{A} \cdot \textrm{P} \cdot \textrm{I}$ and ${\chi _0}$ is the diattenuation axis. Since the optics of DPC is not completely circular symmetry, the ${\varepsilon}$ corresponding to the same field of view angle may be different. Therefore, it is inaccurate to obtain the ${\varepsilon}$ corresponding to each pixel in the full field of view only by the curve of ${\varepsilon}$ fitted from the same azimuth. Instead, the sampling points of the measured ${\varepsilon}$ should increase and be distributed evenly on the whole image surface. Then the measurement results could be interpolated to obtain the corresponding ${\varepsilon}$ of each pixel. Obviously, the more sampling points, the more accurate the final results will be, but the more time-consuming it is. Considering the above factors and the $1024 \times 1024$ effective pixels of DPC, the number of finally selected sampling points is $31 \times 31$. The sampling points in the central field of view contain approximately $11 \times 11$ pixels and more in the edge field of view due to radial distortion. The calibration system for ${\varepsilon}$ consists of a halogen light source with a collimator, several filters, a precise turntable equipped with a polarizer and a two-dimensional turntable, as shown in Fig. 7(a). The sampling points distribution in the full field of view is shown in Fig. 7(b). The light path drawing of DPC diattenuation of the optics calibration system is shown in Fig. 8.

Fig. 7. (a) DPC diattenuation of the optics calibration system. (b) Full field of view sampling points composite image.

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Fig. 8. Light path drawing of DPC diattenuation of the optics calibration system.

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The details of the proposed method for improving the calibration accuracy of the diattenuation of the optics of the DPC are as follows:

1. Remove the rotating wheel of the DPC and place the DPC facing the halogen light source so that it can be imaged within the central field of view of the DPC. A filter which has the same spectral response as the filter on the DPC is placed in front of the light source to limit its spectral range, followed by a precise turntable equipped with a polarizer. The turntable can drive the polarizer and the relative rotation angle can be accurately recorded. The two-dimensional turntable which holds the DPC rotates to make the light source be imaged at the position of the first sampling point of DPC.
2. The turntable drives the polarizer to rotate from 0 to 360 degrees at 15-degree intervals. The light source is continuously imaged multiple times at each rotation angle. Collect the images and perform preprocessing, including multiple image averaging, removal of smearing effect, removal of background signal. Calculate the pixel coordinates of the centroid of each image point and mean DN values of them, then the corresponding ${\varepsilon}$ of the sampling point can be obtained by performing a polynomial fitting with Eq. (10).
3. The two-dimensional turntable rotates to make the light source be imaged at the position of the next sampling point of DPC. Repeat step 2 to calculate the diattenuation of each sampling point of DPC, and then the results are interpolated to obtain the corresponding ${\varepsilon}$ of each pixel.

For example, the diattenuations of optics corresponding to each pixel at 490 nm were obtained by the proposed method, as shown in Fig. 9(a). As a comparison, they were also calculated according to the original method, and the differences between the diattenuations of optics calculated by the two methods are shown in Fig. 9(b). It can be seen from Fig. 9(a) that the ${\varepsilon}$ of DPC is not circularly symmetric, and its shape seems to be the superposition of many concentric ellipses. There are two low value zones in the figure and the minimum value does not appear in the center, which is very similar to the results of 3MI [35]. This is mainly caused by the non-ideality of the processing and adjustment of the optical system and the interaction between the optical system and the coating. As can be seen from Fig. 9(b), the ${\varepsilon}$ obtained by these two methods are quite different, and the maximum difference is about 0.005. The accuracy of ${\varepsilon}$ is determined by the deviation between the fitted value and measured value. Although the number of finally selected sampling points is $31 \times 31$, which makes the uncertainty of ${\varepsilon}$ obtained by interpolation better than ${\pm} 0.2\%$, the ${\varepsilon}$ at the corners can only be obtained by extrapolation, which makes the uncertainty of ${\varepsilon}$ ${\pm} 0.4\%$. For comparison, the uncertainty of ${\varepsilon}$ obtained by the previous method reached ${\pm} 1\%$.

Fig. 9. (a) Diattenuations of optics of 490nm band of the DPC obtained by the proposed method. (b) Differences between the diattenuations of optics calculated by the proposed method and the original method.

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4.2. Absolute azimuth angle

Although only the relative azimuth angle errors between different polarizers cause errors in the DoLP, it can be decreased by improving the accuracy of the absolute azimuth angles of the three channels, thereby improving the polarization measurement accuracy of DPC. The calibration result of the absolute azimuth angle can be improved by measuring a large number of incident light with known polarization state, which can be obtained by a polarizing system. The size of the polarizing system is $600 \times 600 \times 1500\textrm{m}{\textrm{m}^3}$. It consists of an integrating sphere emitting non-polarized radiation and four glass stacks with the size of $600 \times 220 \times 50\textrm{m}{\textrm{m}^3}$, which can produce polarized light with known DoLP and AoLP. The glass stacks, made of K9, have an adjustable range of relative angle from 0 to 65 degrees with the uncertainty of ${\pm} 0.01$ degree. The absolute accuracy of the DoLP of the polarizing system is better than ${\pm} 0.002$ by modelling the response of the glass stacks, which takes into account the accuracy of the relative angle and refractive index of the glass stacks. The accuracy of the AoLP of the polarizing system is better than ${\pm} 0.01$ degree, which depends on the accuracy of the rotation angle of the motor. Because of the calibration error of the relative transmission of the polarized channels at other positions except the central field of view, the incident light can only be imaged in the central field of view of DPC. Let the DoLP of the output light of the polarizing system be set to m, and the measurement results of the output light to $M({\alpha ^1},\textrm{ }{\alpha ^2},\textrm{ }{\alpha ^3})$. When the difference between them is the smallest, the corresponding azimuths are the optimized results. Therefore, the objective function can be established as

(12)$$E({{\alpha^1},\textrm{ }{\alpha^2},\textrm{ }{\alpha^3}} )= {\sum\limits_{i = 1}^n {||{{M_i}({{\alpha^1},\textrm{ }{\alpha^2},\textrm{ }{\alpha^3}} )- {m_i}} ||} ^2}$$

where n is the total number of incident light, ${\alpha ^1}$, ${\alpha ^2}$ and ${\alpha ^3}$ are parameters to be optimized, and the initial values of them can be obtained by combining the Glan-Taylor prism with the straight-edge structure and relative azimuth transfer. The optimization range of each parameter is determined by the initial values and its uncertainty, as shown in Eq. (13).

(13)$$\left\{ {\begin{array}{{l}} {\alpha_0^2 - {U_{{\alpha^2}}} \le {\alpha^2} \le \alpha_0^2 + {U_{{\alpha^2}}}}\\ {{\alpha^2} - \alpha_0^2 + \alpha_0^1 - {U_{{\alpha^1}}} \le {\alpha^1} \le {\alpha^2} - \alpha_0^2 + \alpha_0^1 + {U_{{\alpha^1}}}}\\ {{\alpha^2} - \alpha_0^2 + \alpha_0^3 - {U_{{\alpha^3}}} \le {\alpha^3} \le {\alpha^2} - \alpha_0^2 + \alpha_0^3 + {U_{{\alpha^3}}}} \end{array}} \right.$$

where $\alpha _0^1$, $\alpha _0^2$ and $\alpha _0^3$ are the initial values of ${\alpha ^1}$, ${\alpha ^2}$ and ${\alpha ^3}$, and ${U_{{\alpha ^1}}}$, ${U_{{\alpha ^2}}}$ and ${U_{{\alpha ^3}}}$ the uncertainties of ${\alpha ^1}$, ${\alpha ^2}$ and ${\alpha ^3}$. The calibration system for ${\alpha ^a}$ consists of an integrating sphere with a collimator and polarizing system, as shown in Fig. 10. Its light path drawing is shown in Fig. 11. The details of the proposed method for improving the calibration accuracy of the absolute azimuth angle of the DPC are as follows:

1. Place the DPC on a two-dimensional turntable, facing the light output from the polarizing system, so that it can be imaged in the central field of view of the DPC, as shown in Fig. 10.
2. Adjust the angle of the glass stacks to make the DoLP of the output light vary from 0.1 to 0.6 with the interval of 0.1, and adjust the azimuth angle of the polarizing system to change the AoLP of the output light from 0 to 160 degrees with the interval of 20 degrees. In each state of the output light, DPC collects 20 images.
3. Collect the images and perform preprocessing for the images to eliminate the influence of noise. Calculate the DoLP for each processed image. and the optimized azimuth angle can be obtained by minimizing Eq. (12).

Fig. 10. DPC absolute azimuth angle parameter accuracy improvement system.

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Fig. 11. Light path drawing of DPC absolute azimuth angle parameter accuracy improvement system.

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The initial values and optimization results of the absolute azimuth of the polarized channels for three polarized spectral band are shown in Table 2. The maximum difference between the optimization results and the initial values is 0.96 degree. The residuals of the set DoLPs and calculated DoLPs of the 490nm band of the DPC are shown in Fig. 12, and the other two bands are similar. As can be seen form Fig. 12, the residuals calculated by employing the corrected parameters are significantly lower than that calculated by employing the original parameters. Through modeling analysis and a large number of verification experiments, the relative azimuth angle accuracy of the optimized results of the three channels is better than ${\pm} 0.05$ degree. This is mainly affected by the SNR of DPC and the uncertainty of polarizing system.

Fig. 12. Residuals of the set DoLPs and calculated DoLPs of the 490nm band of the DPC, employing the corrected parameters and the original parameters, respectively.

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Table 2. Initial values and optimization results of the absolute azimuth angle.

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4.3. Relative transmission of the polarized channels

From the analysis of the DPC polarization measurement error in the previous section, it can be seen that the calibration error of the relative transmission is the most important factor affecting the DPC polarization measurement accuracy. However, due to the nonuniformity of the filter, except for the central field of view, the error between the calibration result and the true value of T is large. The solution is to match T with each pixel one by one, that is, to calculate the corresponding T of each pixel instead of the original T, which is only obtained by calculating the ratio of the response of the polarized channels in the central field of view. For polarized bands, when the uniform non-polarized light is incident, the response of the polarization channel of DPC can be obtained from its radiometric model, as shown in Eq. (14).

(14)$$DN_{i,j}^{k,a} = G \cdot t \cdot {A^k} \cdot T_{i,j}^{k,a} \cdot P_{i,j}^k \cdot ({1 + {\varepsilon}_{i,j}^k \cdot \textrm{cos}[2({\alpha^{k,a}} - \phi )]} )\cdot I_{i,j}^k\textrm{ + }C_{i,j}^k$$

where $T_{i,j}^{k,a}$ is the relative transmission corresponding to the pixel $(i,j)$, and can be expressed as

(15)$$\left\{ {\begin{array}{{l}} {T_{i,j}^{k,1} = \frac{{DC_{i,j}^{k,1} \cdot ({1 + {\varepsilon}_{i,j}^k \cdot \textrm{cos}[2({\alpha^{k,2}} - {\phi_{i,j}})]} )}}{{DC_{i,j}^{k,2} \cdot ({1 + {\varepsilon}_{i,j}^k \cdot \textrm{cos}[2({\alpha^{k,1}} - {\phi_{i,j}})]} )}}}\\ {T_{i,j}^{k,2} = 1}\\ {T_{i,j}^{k,3} = \frac{{DC_{i,j}^{k,3} \cdot ({1 + {\varepsilon}_{i,j}^k \cdot \textrm{cos}[2({\alpha^{k,2}} - {\phi_{i,j}})]} )}}{{DC_{i,j}^{k,2} \cdot ({1 + {\varepsilon}_{i,j}^k \cdot \textrm{cos}[2({\alpha^{k,3}} - {\phi_{i,j}})]} )}}} \end{array}} \right.$$

The response to uniform non-polarized light in all field of view can be obtained by an all field of view response acquisition system and the method of regional imaging and stitching. The idea of this method is to divide the image surface of the DPC into N regions. The two-dimensional turntable drives the DPC to rotate, so that the stabilized integrating sphere light source is sequentially imaged on each region to obtain the response of the region, and finally the responses of all regions are stitched to obtain the DPC full field of view response [12]. The relationship between the state of the turntable and the coordinates of the central point of the region in the instrument reference frame is shown in Eq. (16).

(16)$$\left\{ \begin{array}{l} \tan \gamma ={-} y/f\\ \tan \varphi ={-} x/\sqrt {{y^2} + {f^2}} \end{array} \right.$$

where $\gamma$ and $\varphi$ represent the pitching angle and azimuth respectively, $\textrm{(x,y)}$ denotes the coordinates of the central point of one region and $\textrm{f}$ is the focal length of the DPC. The calibration system consists of an integrating sphere light source which provide uniform non-polarized light and a two-dimensional turntable to change the relative position of the light source outlet and DPC, as shown in Fig. 13. The light path drawing of this system is shown in Fig. 14. The integrating sphere light source has been calibrated in National Institute of Metrology (NIM) and its degree of polarization is lower than 0.002.

Fig. 13. DPC all field of view response acquisition system.

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Fig. 14. Light path drawing of DPC all field of view response acquisition system.

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Fig. 15. Relative transmission corresponding to each pixel for 490nm band of the DPC.

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The details of the proposed method for enhancing the accuracy of the relative transmission of the DPC are as follows:

1. Place the DPC on the two-dimensional turntable, adjust the position of the turntable and DPC so that the center of the integrating sphere light source could image on the 0-degree field of view of the DPC when the turntable is in the initial state.
2. Divide the image surface of the DPC into $15 \times 15$ regions, the integrating sphere is successively imaged in each region of the DPC. Collect multiple images and perform preprocessing.
3. The response in all regions are stitched to obtain the DPC full field of view response of each spectral polarized channels, and the corresponding T of each pixel can be obtained by Eq. (15).

For example, the corrected relative transmission corresponding to each pixel for 490nm band of the DPC were obtained by the proposed method, as shown in Fig. 15. As can be seen from Fig. 15 that the values of ${T^{490,1}}$ and ${T^{490,3}}$ are about 0.980 and 0.995 in the central field of view, while their distribution in the full field of view is about 0.974 to 0.996 and 0.985 to 1.01, respectively. The uncertainty of T is better than ${\pm} 0.2\%$, which mainly depends on the stability of the integrating sphere light source. For comparison, the uncertainty of T obtained by the previous method reaches ${\pm} 2\%$ due to the non-uniformity of the filters.

In summary, three simple and practical calibration methods are proposed for DPC parameters in the section. Compared with the initial methods, the accuracy of these parameters has been significantly improved. To facilitate reading and comparison, a brief description of the initial calibration methods and improved calibration methods is shown in Table 3.

Table 3. Brief description of the initial calibration methods and improved calibration methods.

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5. Verification experiments and discussion

The verification experiments were carried out in the laboratory to verify the improvement of the parameters modified by the proposed methods on the polarization measurement accuracy of DPC. The first experiment was based on an integrating sphere non-polarized radiation source and a two-dimensional turntable, which is consistent with the DPC all field of view response acquisition system, as shown in Fig. 13. The purpose is to verify whether the corrected parameters improve the inversion accuracy of non-polarized light for all field of view of DPC. The method of regional imaging and stitching was used for obtaining the response values of all pixels of the three polarized bands. The original parameters and the corrected parameters were used to calculate the DoLP of the preprocessed data, respectively, and the results are shown in Fig. 16, Fig. 17, and Fig. 18.

Fig. 16. Degree of polarization of the non-polarized light measured by each pixel of the 490nm band of the DPC. (a) Employing the original calibration parameters. (b) Employing the corrected calibration parameters.

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Fig. 17. Degree of polarization of the non-polarized light measured by each pixel of the 670nm band of the DPC. (a) Employing the original calibration parameters. (b) Employing the corrected calibration parameters.

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Fig. 18. Degree of polarization of the non-polarized light measured by each pixel of the 865nm band of the DPC. (a) Employing the original calibration parameters. (b) Employing the corrected calibration parameters.

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As can be seen, although the three polarized spectral bands went through the same calibration process, the diattenuation of the optics and the relative transmittance of the filters are different, which leads to a large difference in the inversion results of the three polarized spectral bands for non-polarized light, whether the corrections are performed or not. Especially for the 865nm band, there are a lot of stripes in the Fig. 18, which is mainly because there are similar stripes distribution of the relative transmittance of the filters used in 865 nm band, as shown in Fig. 19.

Fig. 19. Relative transmission corresponding to each pixel for 865nm band of the DPC.

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Compared with the original parameters, the DoLP of non-polarized light calculated with the corrected parameters has a great improvement in all three polarized bands. When the original parameters are employed, the polarization measurement errors of the three polarized bands are all less than 0.016, and the polarization measurement errors in the central field of view are smaller than that in the edge field of view, which is consistent with the theoretical analysis in Section 3. When the corrected parameters are employed, the polarization measurement errors of the three polarized bands are all less than 0.011, and the errors are mainly concentrated in the edge field of view. The larger polarization measurement errors in the corners is mainly caused by the calibration error of ${\varepsilon}$ in the corner, which is not sampled but calculated by extrapolation. The mean values and the root mean square errors (RMSE) of the DoLP of the non-polarized light measured by the three polarized bands of DPC were calculated, as shown in Table 4. The results show that the mean values and RMSEs of the inversion results in all field of view of DPC obtained with the corrected parameters are lower than those obtained with the original parameters.

Table 4. Mean values and RMSEs of the DoLP of the non-polarized light measured by the three polarized bands.

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To verify whether the corrected parameters improve the polarization measurement accuracy of DPC when measuring partially polarized light, the second experiment was carried out. The test system is based on a polarizing system and a two-dimensional turntable, which is same as the DPC absolute azimuth angle parameter accuracy improvement system, as shown in Fig. 10. By changing the angle of the glass stack of the polarizing system, the DoLP of the output partially linearly polarized light was set to 0.1, 0.2, 0.3 and 0.4, respectively. These values were determined by modelling the response of the glass stacks. The uncertainty of the polarizing system is better than ${\pm} 0.002$ in the wavelength range from 400 to 1200 nm. The two-dimensional turntable drove the DPC to rotate and the output light was imaged at several different fields of view angles between -55 degrees and 55 degrees (diagonal direction) of DPC, respectively. The original parameters and the corrected parameters were used to calculate the DoLPs of the partially linearly polarized light with different setting values imaged at different fields of view angles, respectively, and the results of three polarized bands of DPC are shown in Fig. 20, Fig. 21, and Fig. 22.

Fig. 20. Results of DoLPs of the polarized light measured by the 490nm band of the DPC, employing the corrected parameters and original parameters, respectively. The setting values of DoLPs is (a) 0.1, (b) 0.2, (c) 0.3 and (d) 0.4, respectively.

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Fig. 21. Results of DoLPs of the polarized light measured by the 670nm band of the DPC, employing the corrected parameters and original parameters, respectively. The setting values of DoLPs is (a) 0.1, (b) 0.2, (c) 0.3 and (d) 0.4, respectively.

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Fig. 22. Results of DoLPs of the polarized light measured by the 865nm band of the DPC, employing the corrected parameters and original parameters, respectively. The setting values of DoLPs is (a) 0.1, (b) 0.2, (c) 0.3 and (d) 0.4, respectively.

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As can be seen from the figures, compared with the results calculated by the original parameters, the calculated DoLPs of the incident light using the corrected parameters is basically closer to the setting values, and the maximum polarization measurement errors of the three polarization bands reduce from 1.18×10⁻², 7.30×10⁻³, and 8.73×10⁻³ to 3.99×10⁻³, 3.51×10⁻³, and 4.97×10⁻³, respectively. Compared with the maximum polarization measurement errors of 0.009, 0.004 and 0.003 in the three polarized bands which are shown in the early paper [12], the present results are inconsistent with those and seem to be worse. This is because the two test results are from two different DPCs, and their optical systems, coatings and CCD detectors are different. Especially for CCD detectors, the effective pixels of the CCD used by the DPC aboard the GF-5 satellite are $512 \times 512$, while the GF-5 (02) satellite are $1024 \times 1024$. Although this change makes the DPC have higher spatial resolution, the smaller pixel size reduces the SNR of DPC from 500 to about 300, so the maximum polarization measurement errors also increase. The maximum polarization measurement errors of DPC aboard the GF-5 (02) satellite will be very close to the previous results if the $2 \times 2$ pixels merging is performed.

Furthermore, the mean absolute errors (MAE) and RMSEs of the DoLPs of the incident light measured by the three polarized bands were calculated, as shown in Table 5. The results show that both the MAEs and RMSEs of the inversion results of DPC obtained with the corrected parameters are much lower than those obtained with the original parameters. These facts verify the improvement of the DPC polarization measurement accuracy when the parameters are modified by the proposed methods.

Table 5. MAEs, RMSEs and Maximum deviations of the DoLPs of the partially linearly polarized light measured by the three polarized bands.

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6. Conclusion

In this paper, the relationship between the polarization measurement accuracy of DPC and the parameters calibration errors caused by the nonideality of components of DPC is analyzed, and a series of simple and practical methods are proposed to improve the calibration accuracy of the parameters-the diattenuation of the optics, absolute azimuth angle, and relative transmission corresponding to each pixel, thereby improving the polarization measurement accuracy of DPC. The methods include increasing the sampling points and obtaining the diattenuation of the optics corresponding to each pixel by interpolation, establishing the objective equation and obtaining the optimal solution of absolute azimuths by using optimization algorithm, and calculating the relative transmission corresponding to each pixel (a matrix) to replace the original relative transmission corresponding to central field of view (a number). Compared with the original methods, the accuracy of the diattenuation of the optics, relative azimuth angle, and relative transmission of three polarized channels obtained with the improved methods are improved from ${\pm} 1\%$, 0.1 degree and ${\pm} 2\%$ to ${\pm} 0.4\%$, 0.05 degree and ${\pm} 0.2\%$, respectively. To verify the improvement of the parameters modified by the proposed methods on the polarization measurement accuracy of DPC, two verification experiments were carried out. The first experiment was based on an integrating sphere non-polarized radiation source and a two-dimensional turntable to verify whether the corrected parameters improve the inversion accuracy of non-polarized light for all field of view of DPC. The experimental results show that when the corrected parameters were employed, the average error in measuring the degree of linear polarization of non-polarized light source for all pixels in the three polarized bands reduced from 3.95×10⁻³, 1.99×10⁻³ and 2.76×10⁻³ to 1.15×10⁻³, 7.80×10⁻⁴ and 8.76×10⁻⁴, respectively, and the RMSEs also reduced. The second experiment was based on a polarizing system and a two-dimensional turntable to verify whether the corrected parameters improve the polarization measurement accuracy of DPC to partially polarized light. The experimental results show that when the corrected parameters were employed, the maximum deviation of degree of polarization between the setting values of the polarizing system and measured values of the DPC at several different fields of view angles between -55 degrees to 55 degrees for each polarized spectral band reduced from 1.18×10⁻², 7.30×10⁻³, and 8.73×10⁻³ to 3.99×10⁻³, 3.51×10⁻³, and 4.97×10⁻³, respectively. Both the MAEs and the RMSEs of the degree of linear polarization obtained with the corrected parameters are much lower than those obtained with the original parameters. All of these verify the improvement of DPC polarization measurement accuracy when the parameters are modified by the proposed methods. The proposed methods are also applicable to the same type of large field of view polarimetric imager as DPC.

Appendix

The incident light can be decomposed into mutually orthogonal P-light propagating in the tangential direction and S-light propagating in the sagittal direction. When the light slants into the components, the polarization effect is introduced due to the different transmittance of P-light and S-light, which can be expressed as diattenuation and shown in Eq. (17).

(17)$${\varepsilon} \textrm{ = }\frac{{{T_s} - {T_p}}}{{{T_s} + {T_p}}}$$

where ${T_s}$ the transmittance of S-light and ${T_p}$ the transmittance of P-light. The ${\varepsilon}$ increases with the increase of incident angle. The optics of the back group of DPC is shown in Fig. 23. Since the optics is telecentric, the main rays of each beam of the light arriving at the CCD are parallel. The F# of the optics is 4.6, which results in a maximum angle of 6.2 degrees between the main ray and the edge ray. Considering the wedge angle of the wedge prism is 0.1 degree, the maximum angle between the incident light and the normal of the first surface of the wedge prism is 6.3 degrees (${\beta }= 6.3^\circ $).

Fig. 23. Optics of the back group of DPC.

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To keep the polarization state of light unchanged before and after passing through the wedge prism, its surface is coated with multi-layer bandpass dielectric coating. The transmittance of S-light and P-light at each wavelength of each polarized band was simulated when light is incident at 6.3 degrees by using the TFCala software. The results of the three polarized bands of DPC are shown in Fig. 24(a), Fig. 25(a), and Fig. 26(a).

Fig. 24. (a) Simulated results of transmittance of S-light and P-light at each wavelength of 490 nm band when light is incident at 6.3 degrees. (b) Diattenuation of the wedge prism at each wavelength of 490 nm band.

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Fig. 25. (a) Simulated results of transmittance of S-light and P-light at each wavelength of 670 nm band when light is incident at 6.3 degrees. (b) Diattenuation of the wedge prism at each wavelength of 670 nm band.

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Fig. 26. (a) Simulated results of transmittance of S-light and P-light at each wavelength of 865 nm band when light is incident at 6.3 degrees. (b) Diattenuation of the wedge prism at each wavelength of 865 nm band.

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As can be seen, the transmittance of S-light and P-light are very close in both three polarized bands. For example, the transmittance of S-light is 0.997227, and the transmittance of P-light is 0.997233 at a wavelength near490nm. The diattenuation of wedge prism at each wavelength of the three polarized bands were calculated, and the results are shown in Fig. 24(b) to Fig. 26(b). As can be seen, the diattenuation of wedge prism is less than $4 \times {10^{\textrm{ - }5}}$, $8 \times {10^{\textrm{ - }5}}$ and $4 \times {10^{\textrm{ - }5}}$ respectively at the central wavelength of each polarized band, while relatively large at the edge wavelength. Combined with the spectral response function, the diattenuation of the wedge prism of the three bands when light is incident at 6.3 degrees were calculated, as shown in Table 6. Since the calculated results correspond to the maximum incident angle of the light beam, the influence of the wedge prism on the polarization state of incident light is lower than the values and are also far lower than the influences of other components (the specific values are shown in the text). Therefore, the wedge prism can be ignored in the radiometric model.

Table 6. Diattenuation of the wedge prism of the three bands when light is incident at 6.3 degrees.

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Funding

Advanced Polarimetric Remote Sensing Technique and Applications Project (GJTD-2018-15).

Acknowledgments

Chan Huang is very grateful to Miss Ruiqi Hu for her care, encouragement and understanding over the years.

Disclosures

The authors declare no conflicts of interest.

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Pixel	$ε$	$T^{1}$	$T^{2}$	$T^{3}$	$α^{1}$	$α^{2}$	$α^{3}$
A	0.003	0.9821	1	0.9947	60.20	0.93	-60.04
B	0.055	0.9821	1	0.9947	60.20	0.93	-60.04

Spectral band (nm)		Absolute azimuth angle (degree)
Spectral band (nm)		Initial value	Optimization value
490	P1	60.20	59.38
	P2	0.93	−0.03
	P3	−60.04	−61.09
670	P1	60.24	60.83
	P2	0.32	0.85
	P3	−60.21	−59.62
865	P1	61.02	60.72
	P2	0.91	0.54
	P3	−59.46	−59.91

Parameters		Methods	Uncertainty
Initial	$ε$	Measure diattenuation as a function of field of view angle at a single azimuth angle, and apply to all azimuth angles, assuming circular symmetry.	$\pm 1 %$
	$α$	Rotate the Glan-Taylor prism and straight-edge structure to calibrate the absolute azimuth angle of the P2 channel. Rotate a polarizer to obtain the relative azimuth angles of three polarized channels.	The relative azimuth accuracy is better than $\pm 0.1$ degree.
	$T$	Measure the relative response of three polarized channels in the central field of view of DPC.	$\pm 2 %$
Improved	$ε$	Measure diattenuation as a function of field of view angle and azimuth angle.	$\pm 0.4 %$
	$α$	Establish the objective equation and using the optimization algorithm to obtain the optimal solution of absolute azimuth angles.	The relative azimuth accuracy is better than $\pm 0.05$ degree.
	$T$	Measure the relative response of three polarized channels in all field of view of DPC.	$\pm 0.2 %$

Spectral band (nm)	Mean value		RMSE
Spectral band (nm)	Original	Corrected	Original	Corrected
490	3.95×10⁻³	1.15×10⁻³	3.15×10⁻³	1.21×10⁻³
670	1.99×10⁻³	7.80×10⁻⁴	1.51×10⁻³	6.46×10⁻⁴
865	2.76×10⁻³	8.76×10⁻⁴	1.72×10⁻³	8.61×10⁻⁴

Spectral band (nm)	Set value of DoLP	MAE		RMSE		Maximum deviation
Spectral band (nm)	Set value of DoLP	Original	Corrected	Original	Corrected	Original	Corrected
490	0.1	3.21×10⁻³	1.19×10⁻³	1.82×10⁻³	1.41×10⁻³	5.29×10⁻³	3.06×10⁻³
	0.2	3.96×10⁻³	1.40×10⁻³	2.85×10⁻³	1.67×10⁻³	8.02×10⁻³	3.58×10⁻³
	0.3	4.10×10⁻³	1.29×10⁻³	3.79×10⁻³	1.59×10⁻³	1.18×10⁻²	3.92×10⁻³
	0.4	4.98×10⁻³	1.36×10⁻³	3.89×10⁻³	1.62×10⁻³	1.16×10⁻²	3.99×10⁻³
670	0.1	2.91×10⁻³	7.00×10⁻⁴	2.89×10⁻³	7.88×10⁻⁴	7.30×10⁻³	1.94×10⁻³
	0.2	2.93×10⁻³	8.62×10⁻⁴	2.89×10⁻³	1.04×10⁻³	6.43×10⁻³	2.45×10⁻³
	0.3	2.71×10⁻³	9.99×10⁻⁴	2.76×10⁻³	1.18×10⁻³	6.15×10⁻³	3.00×10⁻³
	0.4	2.95×10⁻³	1.17×10⁻³	2.73×10⁻³	1.43×10⁻³	5.94×10⁻³	3.51×10⁻³
865	0.1	1.63×10⁻³	1.09×10⁻³	1.95×10⁻³	8.51×10⁻⁴	4.87×10⁻³	2.93×10⁻³
	0.2	2.38×10⁻³	1.37×10⁻³	2.62×10⁻³	1.22×10⁻³	5.68×10⁻³	4.25×10⁻³
	0.3	3.52×10⁻³	1.70×10⁻³	3.14×10⁻³	1.30×10⁻³	6.75×10⁻³	4.82×10⁻³
	0.4	4.52×10⁻³	2.42×10⁻³	3.86×10⁻³	1.49×10⁻³	8.73×10⁻³	4.97×10⁻³

Polarization measurement accuracy analysis and improvement methods for the directional polarimetric camera

Abstract

1. Introduction

2. DPC instrumental concept and radiometric model

2.1 DPC instrumental concept

2.2 DPC radiometric model

3. DPC polarization measurement accuracy analysis

3.1. Diattenuation of the optics

3.2. Relative transmission of the polarized channels

3.3. Absolute azimuth angle

4. DPC polarization measurement accuracy improvement methods

4.1. Diattenuation of the optics

4.2. Absolute azimuth angle

4.3. Relative transmission of the polarized channels

5. Verification experiments and discussion

6. Conclusion

Appendix

Funding

Acknowledgments

Disclosures

References

Cited By

Figures (26)

Tables (6)

Equations (17)

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