## Abstract

Temporal noise and spatial non-uniformity primarily limit the measurement precision of division of focal plane (DoFP) polarimeters, based on which this study proposes an error model for DoFP polarimeters. The closed-form expressions of the estimation error of the main polarization parameters (Stokes vector, degree of linear polarization, and angle of linear polarization) are derived. Compared with the existing error models for DoFP polarimeters in the presence of temporal noise, the proposed model modifies the normalization condition in traditional calibration methods of DoFP polarimeters and clarifies the selection rule of the coefficient matrix leading to more accurate precision estimation; and experiments using linearly polarized light on a real-world DoFP polarimeter prove its validity.

© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

Polarimeters can record full or partial polarization information on the imaging focal plane array (FPA), using which the main polarization parameters such as the Stokes vector, the degree of linear polarization (DoLP), and the angle of polarization (AoP) of the incident light can be reconstructed. Subsequently, the material, roughness, surface shape, and other information of the target can be deduced. Polarization imaging technology is therefore extensively used in target detection and classification [1–3], three-dimensional shape reconstruction [4–6], space remote sensing detection [7–9], and medical biological imaging [10,11]. To accurately reconstruct the polarization information of the target scene, it is necessary to establish a polarization imaging model based on the imaging process and, thereafter, study the detection limit of the polarization imaging system.

A division of focal plane (DoFP) polarimeter [12–15] employs simultaneous polarization imaging mode. This polarimeter integrates a CCD/CMOS sensor and an aluminum nanowire polarizer filter array with a similar pixel structure, as in the imaging FPA [12]. Each super-pixel is composed of a 2 × 2 neighborhood of pixels (Fig. 1). Although the DoFP polarimeter sacrifices a part of the spatial resolution, the complete target polarization information in the four directions (0°, 45°, 90°, and 135°) can be simultaneously obtained by a super-pixel. Furthermore, the DoFP polarimeter does not contain any moving parts; it is compact and stable, rendering it the best choice for field work. With developments in nanomanufacturing technology, companies such as FLIR, 4D Technology, and LUCID Vision Labs have successively launched DoFP polarimeter products that can be used for precision measurement. This study mainly discusses a method for estimating the errors of DoFP polarimeters.

Temporal noise and spatial non-uniformity are the two primary error sources for DoFP polarimeters. The estimation error of DoFP polarimeters in the presence of temporal noise has been extensively analyzed. However, to the best of our knowledge, the estimation error of DoFP polarimeters originating from spatial non-uniformity has not been publicly reported. In the published literatures [16–18], according to error propagation theory [19], the estimation error of the main polarization parameters in the presence of temporal noise (primarily readout and photon shot noises) can be calculated using the covariance matrix of the output grayscale caused by the temporal noise coupled with the coefficient matrix characterizing the characteristics of the pixeled polarizer array. Till date, the coefficient matrix of the pixeled polarizer array is typically calculated using the output grayscale through calibration methods [20–22]. However, the spatial distribution characteristics of the pixeled polarizers result in an inability to completely eliminate the influence of the spatial non-uniformity of the sensor during the calibration processes. When analyzing the influence of temporal noise, it should be statistically independent of spatial non-uniformity; however, because the coefficient matrix introduces part of the spatial non-uniformity of the sensor, the calculated estimation error of the DoFP polarimeters caused by the temporal noise is affected by the spatial non-uniformity of the sensor. This leads to a difficulty in analyzing the temporal noise and spatial non-uniformity independently. Further, when analyzing the influence of spatial non-uniformity, the spatial non-uniformity of the sensor should be completely reflected in the covariance matrix of the output grayscale. However, because the calibrated coefficient matrix retains the influence of the spatial non-uniformity of the sensor, the calculated estimation error of DoFP polarimeters caused by spatial non-uniformity may deviate from the ground truth.

Thus, based on the error propagation theory, we studied the temporal noise (readout and photon shot noises) and the spatial non-uniformity of DoFP polarimeters. Thereafter, an error model of the DoFP polarimeter was established. Subsequently, the closed-form expressions of the estimation error of the main polarization parameters (Stokes vector, DoLP, and AoP) were derived. Compared to models that only analyze the temporal noise of DoFP polarimeters [16–18], our study has achieved that: 1) We have modified the normalization condition in traditional calibration methods of DoFP polarimeters, which results in the calibrated normalized coefficient matrix being independent of the spatial non-uniformity of the sensor and thus truly characterizes the modulation effect of the pixelated polarizer array on the incident light. 2) Targeting the problem that the coefficient matrix of a single super-pixel cannot represent the overall distribution of the coefficient matrix of each super-pixel on the FPA we have clarified the selection rule of the coefficient matrix to obtain a more accurate precision estimation of the DoFP polarimeters. The results of this study can be used to guide the selection of device parameters in the design process and the precision estimation and parameter adjustment in actual measurements for DoFP polarimeters.

The remainder of this paper is organized as follows: Section 2 describes the Stokes vector polarization theory and calibration method of DoFP polarimeters. Section 3 presents the proposed error model of the DoFP polarimeters in detail. Section 4 presents the simulation and analysis of the influence of the camera device parameters and the intensity and polarization statement of the incident light on the measurement precision of DoFP polarimeters. Further, the verified experiments of linearly polarized light and an explanation of the coefficient matrix selection rule are also reported in this section. Finally, Section 6 concludes the study.

## 2. Basic theory of polarization imaging

#### 2.1 Theory of the Stokes vector

The Stokes vector **S** [23] is typically used to describe the polarization characteristics of any light field and can be defined as:

*S*

_{0}is the total light intensity,

*S*

_{1}is the horizontal or vertical linear polarization component,

*S*

_{2}is the linear polarization component of +45° or −45° polarization directions, and

*S*

_{3}is the left- or right-handed circular polarization component. Because the circular polarization component in natural scene radiation is extremely small,

*S*

_{3}is typically considered to be 0. Moreover, DoFP polarimeters respond only to linear Stokes parameters (that is,

*S*

_{0},

*S*

_{1}, and

*S*

_{2}). Thus,

*S*

_{3}was omitted from the Stokes vector mentioned in this study.

DoLP and AoP are typically used to investigate the polarization states of the target scene. DoLP represents the proportion of the linearly polarized component to the total intensity of the light source, while AoP represents the angle between the polarization direction of the maximum incident light energy and the *x*-axis in the reference coordinate system. DoLP and AoP can be calculated using the Stokes vector as follows:

*P*represents DoLP.

*P*∈[0, 1], and

*P*= 1 for linearly polarized light.

*α*represents the AoP.

With the incident Stokes vector **S** = *S*_{0}[1 *P*cos (2*α*) *P*sin (2*α*)] ^{T}, the output grayscale *DN _{mn}* (with the dark offset removed) of a single pixel of DoFP polarimeters can be expressed as:

*m*and

*n*are the pixel coordinates;

*m*∈[1, 2

*M*],

*n*∈[1, 2

*N*]; 2

*M*× 2

*N*is the size of the sensor;

*g*, in units of

*DN*/

*e*

^{-}, is the total gain of the sensor;

*η*, in units of

*e*

^{-}/

*p*

^{∼}, is the quantum efficiency of the sensor;

**w**

*is the coefficient matrix of the pixelated polarizer (*

_{mn}*m*,

*n*), which characterizes the modulation effect of the pixelated polarizer on the incident Stokes vector;

*S*

_{0}, in the unit of

*p*

^{∼}, is the total number of the incident photons captured by the sensor during the integration time

*t*(in the unit of

*s*).

Furthermore, the imaging sensors of the DoFP polarimeters respond only to light intensity. Thus, the pixelated polarizer array coupled in front of the sensor acts as an analyzer. The coefficient matrix **w*** _{mn}* characterizing the effect of this polarizer on the incident Stokes vector is given by:

*τ*and

_{mn}*q*are the transmittance and diattenuation coefficients of the pixelated polarizer at (

_{mn}*m*,

*n*), respectively. Further, the relationship between the diattenuation coefficient

*q*and extinction ratio

_{mn}*ε*

^{2}

_{mn}at (

*m*,

*n*) can be defined as

*q*= (

_{mn }*ε*

^{2}

_{mn}-1)/(

*ε*

^{2}

_{mn}+1). The diattenuation coefficient

*q*varies between 0 and 1, and equals 1 if the polarizer is ideal, while

*θ*is the main direction of the pixelated polarizer at (

_{mn}*m*,

*n*).

#### 2.2 Calibration method of DoFP polarimeters

From Eq. (3), it is evident that the output grayscale error of the DoFP polarimeters is affected by the imaging sensor and the front-mounted polarizer. Typically, the non-ideality of the pixelated polarizer array of DoFP polarimeters can be corrected using calibration methods; therefore, we first accurately calibrate the coefficient matrix of each pixelated polarizer array unit. The adoption of a relatively mature nanomanufacturing technique still results in severe deviation of the optical characteristics of the pixelated polarizer array (caused by manufacturing error) from the ideal situation. Further, as the non-uniformity of the optical characteristics of pixelated polarizers on the entire FPA can occasionally be as high as 20% [20,24,25], calibrating the pixelated polarizer array must be completed first before discussing the estimation error of the DoFP polarimeters.

However, the characteristic parameters of the nanometer-sized pixelated polarizer can only be calculated indirectly through the output grayscale of DoFP polarimeters because of the integration of this array into the sensor. Thus, the super-pixel calibration method [20] is used to obtain the coefficient matrix of the pixelated polarizer array of DoFP polarimeters and this calibration can be regarded as the inverse of the imaging process, as shown in Eq. (3). For the known 3×*K* (*K* is a positive integer) incident Stokes vector matrix ${\mathbf S}_{cal} = [{\mathbf S}_{cal}^1 \cdots {\mathbf S}_{cal}^k ]$, the coefficient matrix **w*** _{mn}* of a single pixel can be calculated as follows:

**DN**

*is the output grayscale vector of the pixel located at (*

_{calmn}*m*,

*n*) and 1×

*K*matrix

**DN**

_{calmn}= ${\mathbf DN}_{calm} = [{\mathrm DN}_{mn}^1 \cdots {\mathrm DN}_{mn}^k ]\,$. ${\mathbf S}_{cal}^\dagger$ is the pseudo-inverse matrix [26] of

**S**

*, and ${\mathbf S}_{cal}^\dagger = \textrm{ }{({\mathbf S}_{cal}^T \cdot {{\mathbf S}_{cal}})^{ - 1}}{\mathbf S}_{cal}^T$.*

_{cal}We assume that the incident Stokes vector **S*** _{cal}* in Eq. (5) is uniform in both the temporal and spatial domain. However, due to the systematic error, the output grayscale

**DN**

*exhibits non-uniformity in space (i.e. spatial non-uniformity) and fluctuates randomly with time (i.e. temporal noise). Typically, the optical performance of a polarizer does not change with time during one imaging process [20]; therefore, the temporal fluctuation of the output grayscale originates entirely from the temporal noise of the sensor. However, when calibrating the coefficient matrix of the pixelated polarizer array, the effect of this temporal noise on the calibration result can be approximately eliminated by averaging multiple image frames in different time series, typically requiring frames greater than 400 [27]. In contrast, the spatial non-uniformity of the output grayscale of the DoFP polarimeters originates from the combined influence of the sensors and the pixelated polarizer array. Unfortunately, the distribution of the spatial non-uniformity of the sensor is generally unknown; thus, it is difficult to obtain an accurate coefficient matrix of a pixelated polarizer array using traditional calibration methods.*

_{calmn}Fortunately, from Eq. (4), we observed that the characteristic parameters *ε* ^{2} and *θ* of the polarizers are related only to the relative magnitudes of the three components of **w*** _{mn}*. This implies that the calibrated values of

*ε*

^{2}and

*θ*are not affected by the spatial non-uniformity of the sensor. We substitute Eq. (4) into Eq. (5), extract the transmittance coefficient τ

*of the pixelated polarizer on the left-hand side, and then move it to the right-hand side of the resulting equation. The calibrated normalized coefficient matrix,*

_{mn}**m**

*can thereafter be calculated as:*

_{mn},*= 2*

_{mn }**w**

*(1) and*

_{mn}**w**

*(1) is the first elements of the coefficient matrix*

_{mn}**w**

*(Eq. (4)).*

_{mn}It is evident from Eq. (6) that the calibrated normalized coefficient matrix **m*** _{mn}* can purely characterize the modulation effect of a pixelated polarizer array on incident light without being affected by the noise of the sensor.

Further, when the pixelated polarizer array of DoFP polarimeters satisfies the assumption of ideal polarizers (that is, the diattenuation coefficients *q* of the four pixels in each super-pixel equal 1, and polarization direction *θ* is equal to 0°, 45°, 90°, and 135°, respectively), the ideal normalized coefficient matrix **M*** _{ideal}* of a single super-pixel can be represented as follows:

**, m**

*, the imaging process described by Eq. (3) can be reformulated as: where*

_{mn}*τ*is the average transmittance coefficient across the pixelated polarizer array, $\tau = \sum\limits_{m = 1}^M {\sum\limits_{n = 1}^N {{\tau _{mn}}} } $.

Therefore, based on the above analysis, the spatial non-uniformity of the DoFP polarimeters mentioned in this study refers to the comprehensive influence of the spatial non-uniformity of the sensor and transmittance coefficient of the pixelated polarizer array.

In this study, the pseudo-inverse estimation method was used to reconstruct the incident Stokes vector. This estimator is optimal in the maximum-likelihood sense when the noise that perturbs the measurement vector **DN** (Eq. (9)) is an additive, white, and Gaussian [26]. The pseudo-inverse estimation method is also a simple closed-form algorithm in the presence of photon shot noise, and provides good results in practice. Moreover, it has been shown in [28] that with a correctly balanced measurement matrix, the pseudo-inverse estimator leads to an estimation variance that is approximately equal to that of the maximum-likelihood estimator. Thus, for each super-pixel, the process of polarization imaging and that of reconstructing the incident Stokes vector using the output grayscale can be represented as:

**DN**

*is the output grayscale vector of the super-pixel and*

_{mn}**DN**

_{mn}**= [**

*DN*

_{mn}DN_{m}_{(n+1)}

*DN*

_{(m+1)(n+1)}

*DN*

_{(m+1)n}]

^{T}.

**M**

*is the coefficient matrix of the super-pixel,*

_{mn}**M**

_{mn}**= [**

**m**

_{mn}**m**

_{m}_{(n+1)}

**m**

_{(m+1)(n+1)}

**m**

_{(m+1)n}]

^{T}and ${{\mathbf M}^\dagger }$ is the pseudo-inverse matrix of

**M**, ${{\mathbf M}^\dagger } = {({{\mathbf M}^T} \cdot {\mathbf M})^{ - 1}}{{\mathbf M}^T}$.

## 3. Estimation error of the Stokes vector

To understand the sensor noise, we referred to the comprehensive noise model [29,30] based on the physical process of photon sampling and quantification in a CCD sensor imaging process (Fig. 2). This model is also applicable to CMOS sensors. Moreover, the primary noise sources for monochrome CCD/CMOS sensors (Table 1) are temporal noise (including readout noise *N _{read}* and photon shot noise

*SN*) and spatial non-uniformity (including photon response non-uniformity

_{ph}*PRNU*).

Furthermore, the estimation error of the Stokes vector can be represented by a covariance matrix, wherein the diagonal elements of the covariance matrix are the estimation variances of the three Stokes parameters. Using the error propagation theory coupled with Eq. (9), the covariance matrix **Γ ^{S}** of the Stokes vector

**S**is:

**Γ**is the covariance matrix of the output grayscale vector

^{DN}**DN**and is related to the noise level of the sensor.

#### 3.1 Temporal noise

Temporal noise is the random fluctuation of the response of a single super-pixel on the FPA in the temporal domain. Therefore, the derivation process of the covariance matrix described in this section is only targeted at a single super-pixel of DoFP polarimeters, where **M** refers to the coefficient matrix of the corresponding super-pixel. To simplify the description, the subscript “*mn*” is omitted in Section 3.1.

### 3.1.1 Readout noise *N*_{read}

_{read}

The readout noise causes the output grayscale to change only in the temporal domain and includes pixel reset noise, thermal noise sources (Johnson-Nyquist), and other minor contributors, such as, frequency-dependent 1/*f* (flicker) noise sources and conductor shot noise [29]. In addition, thermal noise is independent of illumination and occurs regardless of the applied voltage [32]. Further, the level of 1/*f* noise in a CCD sensor is dependent on the pixel sampling rate and a constant operating temperature readout rate of the sensors results in the readout noise being constant.

The readout noise *N _{read}* can be regarded as Gaussian white noise with a mean value of 0 and a variance of

*σ*

^{2}

_{r}(in the unit of

*DN*

^{2}). Considering the statistical properties of the readout noise,

*N*is independent of multiple measurements. Therefore, the covariance matrix

_{read}**Γ**of the readout noise

^{DNr}*N*is expressed as the following diagonal matrix:

_{read}**Γ**of the Stokes vector under the influence of the readout noise

^{Sr}*N*as follows:

_{read}*i*” and “

*j*” are the row or column indices, respectively, in matrix

**Γ**with

^{Sr}*i*,

*j*= 1, 2, 3. ${{\mathbf d}_{ij}} = \textrm{ }{[{{{({{{\mathbf M}^T}{\mathbf M}} )}^\dagger }} ]_{ij}}$.

Thereafter, substituting Eq. (7) into Eq. (12), we obtain the covariance matrix **Γ ^{Sri}** of the Stokes vector by applying the ideal coefficient matrix

**M**

*as follows:*

_{ideal}*N*, is related only to the device parameters (gain

_{read}*g*and quantum efficiency

*η*), the noise level of the sensor, and the transmittance coefficient

*τ*of the pixelated polarizer array. Moreover, this estimation error was not influenced by the intensity and polarization states of the incident light.

### 3.1.2 Photon shot noise *SN*_{ph}

_{ph}

Capturing of photons is a Poisson process for CCD/CMOS sensors [33], arising from random fluctuations in the temporal domain when sampling discrete quanta. In addition, for a single pixel on the CCD/CMOS FPA, the number of photons collected during the integration time follows a Poisson distribution in the temporal domain. Further, the signal standard deviation caused by the photon shot noise is numerically equal to the square root of the mean value of the signal (in units of *e*^{-}).

Under a typical incident light intensity level, the photon shot noise *SN _{ph}* is one of the principal noise sources of the sensor. Considering the nature of this noise, the fluctuation from one intensity measurement to another is linearly independent. Therefore, the covariance matrix

**Γ**of the photon shot noise

^{DNph}*SN*can be calculated using the following diagonal matrix:

_{ph}*i*,” “

*j*,” and “

*t*” are the row or column indices in matrix

**Γ**,

^{DNph}**DN**, or

**M**, and

*i*,

*j*= 1, 2, 3, 4. 〈

**DN**

*〉 is the average output grayscale of the super-pixel in the temporal domain, and $\left\langle {{\mathbf D}{{\mathbf N}_i}} \right\rangle = \sum\limits_{t = 1}^T {{\mathbf D}{{\mathbf N}_i}} (t )$, while*

_{i}*T*is the number of image frames in different time series.

Substituting Eq. (14) into Eq. (10), we obtain the covariance matrix **Γ ^{Sph}** of the Stokes vector under the influence of the photon shot noise

*SN*.

_{ph}*i*,” “

*j*,” “

*l*,” and “

*t*” are the row or column indices in matrix

**Γ**,

^{Sph}**M**, and ${{\mathbf M}^\dagger }$, with

*i*,

*j*= 1, 2, 3 and $\gamma _{ij}^t\textrm{ = }\sum\limits_{l = 1}^4 {{\mathbf M}_{il}^\dagger {\mathbf M}_{jl}^\dagger {\mathbf M}_{lt}^{}}$.

Further, substituting Eq. (7) into Eq. (15), we obtain the covariance matrix **Γ ^{Sphi}** of the Stokes vector by applying the ideal coefficient matrix

**M**

*as follows:*

_{ideal}*SN*is not only related to the quantum efficiency

_{ph}*η*of the sensor and transmittance coefficient

*τ*of pixelated polarizers, but is also affected by the intensity and polarization states of the incident light.

#### 3.2 Spatial non-uniformity

Photon response non-uniformity (*PRNU*) describes the difference in the response of pixels to uniform light sources. *PRNU* originates from variations in the pixel geometry, substrate material, and microlens across the FPA [29] and is a major source of spatial non-uniformity of sensors. Because *PRNU* is caused by the physical properties of sensors, it cannot be eliminated and is therefore typically considered as a normal characteristic of the sensor array used in any polarimeter. Further *PRNU* is a type of multiplicative spatial noise proportional to the incident light intensity and its influence is prominent at high incident intensity levels [34].

The *PRNU* standard deviation *σ _{pr}* (in the

*DN*unit) is proportional to the mean value of the signal. Moreover, according to the statistical properties of

*PRNU*, this noise is independent of multiple measurements. Therefore, its covariance matrix

**Γ**can be calculated using the following diagonal matrix:

^{DNpr}*i*,” “

*j*,” “

*t*,” and “

*w*” are the row or column indices in matrix

**Γ**,

^{DNpr}**DN**, or $\overline {\mathbf {\rm M}}$, with

*i*,

*j*= 1, 2, 3, 4. $\overline {\left\langle {{\mathbf D}{{\mathbf N}_i}} \right\rangle } $ is the average output grayscale of the super-pixel in the temporal-spatial domain and $\overline {\left\langle {{\mathbf D}{{\mathbf N}_i}} \right\rangle } \textrm{ = }\sum\limits_{m = 1}^M {\sum\limits_{n = 1}^N {\sum\limits_{t = 1}^T {{\mathbf D}{{\mathbf N}_i}({m,n,t} )} } }$. Further, $\overline {\mathbf {\rm M}}$ is the mean value of the coefficient matrix of the pixelated polarizer array $\overline {\mathbf M} \textrm{ = }\sum\limits_{m = 1}^M {\sum\limits_{n = 1}^N {{\mathbf M}({m,n} )} }$.

Thus, substituting Eq. (17) into Eq. (10), we obtain the covariance matrix **Γ ^{Spr}** of the Stokes vector under the influence of the

*PRNU*

*i*,” “

*j*,” “

*l*,” “

*t*,” and “

*w*” are the row or column indices in matrix

**Γ**, $\overline {\mathbf {\rm M}}$ or ${\overline {\mathbf {\rm M}} ^\dagger }$, with

^{Spr}*i*,

*j*=1, 2, 3, $\beta _{ij}^t\textrm{ = }\sum\limits_{l = 1}^4 {\overline {\mathbf M} _{il}^\dagger \overline {\mathbf M} _{jl}^\dagger \overline {\mathbf M} _{lt}^2}$, $\chi _{ij}^{tw}\textrm{ = }\sum\limits_{l = 1}^4 {\overline {\mathbf M} _{il}^\dagger \overline {\mathbf M} _{jl}^\dagger {{\overline {\mathbf M} }_{lt}}} {\overline {\mathbf M} _{lw}}$, and ${\overline {\mathbf {\rm M}} ^\dagger }$ is the pseudo-inverse matrix of $\overline {\mathbf {\rm M}}$.

Therefore, substituting Eq. (7) into Eq. (18), we obtain the covariance matrix **Γ ^{Spri}** of the Stokes vector by applying the ideal coefficient matrix

**M**

*as follows:*

_{ideal}*PRNU*is not only related to the noise level of the sensor, but is also affected by the intensity and polarization states of the incident light.

#### 3.3 Estimation error of the polarization parameters

The DoLP and AoP are two important parameters for investigating the polarization states of the target scene as the variances of these parameters are important indices for evaluating the measurement precision of DoFP polarimeters. From Eq. (2), it is evident that DoLP and AoP are nonlinear functions of the three Stokes parameters. Thus, according to error propagation theory, the variances of the DoLP and AoP can be calculated using the covariance matrix **Γ ^{S}** of the Stokes vector caused by temporal noise and spatial non-uniformity. Further, the temporal noise and spatial non-uniformity of the CCD/CMOS sensors are statistically independent, and typically these noises are considered sufficiently small relative to the output grayscale of DoFP polarimeters. Therefore, the covariance matrix

**Γ**of the Stokes vector caused by temporal noise and spatial non-uniformity can be expressed as the sum of the covariance matrices

^{S}**Γ**,

^{Sr}**Γ**, and

^{Sph}**Γ**under the influence of

^{Spr}*N*,

_{read}*SN*, and

_{ph}*PRNU*, respectively [17]:

**Γ**of the Stokes vector caused by temporal noise and spatial non-uniformity. The diagonal elements of

^{S}**Γ**are the estimation errors of the three linear Stokes parameters of DoFP polarimeters caused by temporal noise and spatial non-uniformity. Further it can be easily proven that for an ideal sensor free from spatial non-uniformity, it is unnecessary to perform normalization in the calibration of the coefficient matrix. Therefore, Eq. (20) can be simplified as an error model that considers only the temporal noise of the DoFP polarimeters that is consistent with existing temporal noise models in the literatures [16–18]. Therefore, the existing models that analyze only the temporal noise can be regarded as a special case of the proposed model under the assumption of an ideal sensor (that is, the sensor is free from spatial non-uniformity).

^{S}### 3.3.1 Estimation error of the DoLP

DoLP is a nonlinear function of the three Stokes parameters *S*_{0}, *S*_{1}, and *S*_{2}. The gradient of DoLP can be obtained using Eq. (2) as:

*w*

_{1}= (

*gητ*)

^{-2}

*σ*

^{2}

_{r}and

*w*

_{2}= (2

*ητ*)

^{-1}.

### 3.3.2 Estimation error of the AoP

AoP is a nonlinear function of the Stokes parameters *S*_{1} and *S*_{2}. The gradient of AoP can be obtained using Eq. (2) as:

*w*

_{4}= (4

*ητ*)

^{-1}.

## 4. Result and discussion

#### 4.1 Influence of the device parameters on the measurement precision of DoFP polarimeters

The hardware performance of the sensors and external measurement conditions are the internal and external factors that affect the measurement precision of DoFP polarimeters. Thus, an analysis of these factors can be used to guide the selection of device parameters in the design process, precision estimation, and parameter adjustment in actual measurements. For the former, the polarization parameter estimation error of DoFP polarimeters is related to the noise level of the sensor (i.e. readout noise *N _{read}* and spatial non-uniformity coefficient

*PRNU*), gain

*g*, product of the quantum efficiency

*η*of the sensor, and transmittance coefficients

*τ*of the pixelated polarizer (Eqs. (11)–(25)). In this section, we analyze the impact of the device parameters on the measurement precision of DoFP polarimeters through a simulation.

The variation in the standard deviations of the DoLP and AoP with the read noise *N _{read}* and

*PRNU*coefficient is shown in Fig. 3. When the

*PRNU*coefficient is greater than 0.5% (typically, the real-world

*PRNU*coefficient of the sensor is greater than this value), the influence of spatial non-uniformity on the standard deviations of the DoLP and AoP have the same order of magnitude as that of temporal noise. For

*PRNU*coefficient greater than 1.6%, the influence of spatial non-uniformity on the standard deviations of the DoLP and AoP is greater than that of the temporal noise (

*S*

_{0in}>1 × 10

^{4}

*p*

^{∼}). This shows that it is necessary to investigate the influence of spatial non-uniformity on the estimation error of DoFP polarimeters. When detecting weakly polarized targets with sufficient illumination, DoFP polarimeters with low spatial non-uniformity can achieve higher measurement precision. While when detecting strongly polarized targets in low-illuminance environments, the influence of the readout noise of the sensor is more obvious. In the design process of the DoFP polarimeters, the proposed model can be used to determine the range of the noise level of sensors that meets the measurement precision requirements for different application scenarios.

The variation in the standard deviation of the DoLP and AoP with *η*·*τ* is shown in Fig. 4. For the general real-world sensors (*η*·*τ* greater than 0.1), the influence of spatial non-uniformity on the standard deviations of the DoLP and AoP has the same order of magnitude as that of temporal noise. This shows that it is necessary to investigate the influence of spatial non-uniformity. Based on the existing nanomanufacturing techniques, the quantum efficiency and transmittance of the system can be increased by coating the antireflection film, coupling the microlens array, and selecting an appropriate wave band (*η*·*τ* is a function of wavelength) in the imaging process. Therefore, these methods can effectively improve the measurement precision of DoFP polarimeters.

#### 4.2 Influence of the noise of the sensor on the measurement precision of DoFP polarimeters in imaging process

In the actual imaging process, the measurement precision of DoFP polarimeters is also affected by external conditions, such as the intensity and polarization states of the incident light. In this section, we numerically simulated the influence of the external conditions on the measurement precision of real-world DoFP polarimeters with fixed device parameters and experimentally verified them using linearly polarized light.

### 4.2.1 Numerical simulation

We performed a numerical simulation of the LUCID DoFP polarimeter. The primary parameters of the polarimeter are listed in Table 2. The variation in the standard deviations of the DoLP and AoP with the intensity *S*_{0in}, the DoLP *P _{in}* and the AoP

*α*of the incident light is shown in Figs. 5 and 6. The intensity

_{in}*S*

_{0in}and the DoLP

*P*of the incident light are the primary factors that affect the standard deviations of the DoLP and AoP caused by temporal noise. Thus, appropriately increasing the integration time during measurement to increase the number of incident photons in a single image is an effective method for improving the measurement precision of DoFP polarimeters. The DoLP

_{in}*P*and AoP

_{in}*α*of the incident light are the primary factors that affect the standard deviations of the DoLP and AoP caused by spatial non-uniformity. Rotating the system during measurement to avoid coincidence of the AoP of the incident light with the analyzer directions can lead to high measurement precision of the DoLP and AoP being obtained simultaneously. Comparison between the estimation error of DoFP polarimeters caused by temporal noise and that by spatial non-uniformity shows that when

_{in}*S*

_{0in}is greater than 1.5 × 10

^{3}

*p*, the influence of spatial non-uniformity on the standard deviations of the DoLP and AoP has the same order of magnitude as that of temporal noise. This indicates that it is necessary to analyze the temporal noise and spatial non-uniformity synthetically for DoFP polarimeters. In the actual imaging process of the DoFP polarimeters, the proposed model can be used to guide the adjustment of the system parameter and attitude in order to effectively improve the measurement precision.

^{∼}### 4.2.2 Linearly polarized light experiments

The validity of the above numerical simulation was proven by experiments using linearly polarized light. The configuration of the experimental system is illustrated in Fig. 7. An ideal light source emits uniform and collimated linearly polarized light, however, in practice, it is difficult to obtain such a source. The uniformity rather than the collimation of the light source should be guaranteed firstly because the inter-pixel crosstalk introduced by the non-collimated light incident on the sensors can be effectively calibrated during the calibration process (with the same F number) for DoFP polarimeters [20], while the uniformity of the incident light on the FPA directly affects the accuracy of the experimental results. Therefore, we chose an integrating sphere combined with a THORLABS LPVISC100 polarizer as the polarization light source in the verification experiment, which produced uniform incident linearly polarized light at 603 *±* 10* *nm. Moreover, the influence of the non-ideality of the THORLABS polarizer on the experimental results was calibrated [22]. Further, in the experiment, no lens was installed in front of the DoFP polarimeter to focus on the discussion of the inherent noise characteristics of the sensor [35].

The signal-to-noise ratio (*SNR*) is used as the standard for evaluating the estimation error of the Stokes vector. *SNR* is defined as:

*μ*is the mean value of the signal, and

_{signal}*σ*is the standard deviation of the noise.

_{noise}To verify the validity of the proposed model when the intensity of the incident linearly polarized light changes, we arbitrarily set the incident AoP *α _{in}* to 65° and thereafter, adjusted the iris diaphragm to collect 55 groups of LUCID DoFP polarimeter images with different incident light intensities,

*S*

_{0in}. Subsequently, a total of 500 frames of images in different time series at each incident light intensity were collected. Further, to verify the validity of the proposed model when the AoP of the incident linearly polarized light changes, we arbitrarily set the incident light intensity

*S*

_{0in}to 2.66 × 10

^{4}

*p*

^{∼}and rotated the polarizer to collect 60 groups of LUCID DoFP polarimeter images with the incident AoP

*α*ranging between 0° and 180° (with a step length of 3°). Thus, a total of 500 frames of images in different time series at each incident AoP

_{in}*α*were collected. The sub-region selected was of 300 × 300 pixels (1.1 mm

_{in}^{2}) at the center of the images to maximize the uniformity of the incident light. The variation in the estimation error (

*SNR*s for the Stokes vector and standard deviations of the DoLP and AoP) caused by the temporal noise and spatial non-uniformity with

*S*

_{0in}and

*α*were calculated using the proposed model and the real images (Figs. (8)–(10)).

_{in}Thus, from Fig. 8, the following can be observed: (1) For a certain polarization state of the incident light (Figs. 8(a)–8(c)), the *SNR*s for the three Stokes parameters (*S*_{0}, *S*_{1}, and *S*_{2}) caused by the temporal and total noises increase with the increase in *S*_{0in}, whereas those caused by the spatial non-uniformity are almost independent of *S*_{0in}. (2) For a certain *S*_{0in} (Figs. 8(d)–8(f)), an increase in *α _{in}* results in

*SNR*s for

*S*

_{1}and

*S*

_{2}approximately conforming to the cosine variation with a period of 90°. Further, the experimental results of the Stokes vector on a real-world DoFP polarimeter are highly consistent with the theoretical results calculated by the proposed model. This shows that compared with the model that considers only the temporal noise, applying our model results in a more accurate estimation of the Stokes vector for DoFP polarimeters.

Considering Figs. 9 and 10, the following can be observed: (1) For a certain polarization state of the incident light (Figs. 9(a) and 10(a)), the standard deviations of the DoLP and AoP caused by the temporal and total noises decrease with the increase in *S*_{0in}, whereas those caused by the spatial non-uniformity are almost independent of *S*_{0in}. (2) For a certain *S*_{0in} (as shown in Figs. 9(b) and 10(b)), an increase in *α _{in}*, results in the standard deviations of the DoLP and AoP caused by the spatial non-uniformity and total noise approximately conforming to the cosine variation with a period of 45° (Eq. (23)). However, the deviations caused by the temporal noise fluctuate slightly with the increase in

*α*under similar incident light conditions. Furthermore, the experimental results of the DoLP and AoP on a real-world DoFP polarimeter are highly consistent with the theoretical results calculated by the proposed model, which indicates that compared with the model that considers only the temporal noise, applying our model results in a more accurate estimation of the DoLP and AoP for DoFP polarimeters.

_{in}#### 4.3 Selection rule of the coefficient matrix

The selection rule of the coefficient matrix and its advantages are discussed in this section. When analyzing the temporal noise for DoFP polarimeters, because of the stable optical characteristics of the pixelated polarizer array (the optical performance of the polarizer does not change with time during one imaging process [20]), we directly selected the coefficient matrix **M*** _{mn}* corresponding to the target super-pixel (

*m*,

*n*) to calculate the covariance matrices of the Stokes vector,

**Γ**(

^{Sr}*m*,

*n*) and

**Γ**(

^{Sph}*m*,

*n*), caused by the readout and photon shot noises. Thereafter,

**Γ**(Eq. (20)) caused by the system temporal noise can be obtained by using the average value of each super-pixel across the FPA. In contrast, when analyzing the spatial non-uniformity of the DoFP polarimeters, the coefficient matrix corresponding to each super-pixel on the FPA varies with the spatial position. Therefore, the coefficient matrix of a single super-pixel is unable to represent the entire distribution of the coefficient matrix of the super-pixels on the FPA. Therefore, we select the average coefficient matrix $\overline {\mathbf {\rm M}}$ (Eq. (17)) to calculate

^{St}**Γ**caused by the system spatial non-uniformity in this study.

^{Spr}The influence of selecting the average coefficient matrix $\overline {\mathbf {\rm M}}$ on the estimation precision of **Γ ^{Spr}** under different incident light conditions is discussed below. The maximum relative deviation

*Error*is defined to evaluate the rationality of selecting the average coefficient matrix $\overline {\mathbf {\rm M}}$ in our model as follows:

_{max}*std*represents the ground truth of the corresponding standard deviation.

_{truth}*Error _{max}* represents the maximum deviation, originating from the proposed selection rule of the coefficient matrix, of the estimation error of DoFP polarimeters caused by the spatial non-uniformity from the ground truth under various incident light conditions; which implies that the maximum error may be introduced by the selection rule of the coefficient matrix in this study. For a group of incident linearly polarized light with different polarization states (incident light intensity

*S*

_{0in}

*= 2.66 × 10*

_{ }^{4}

*p*

^{∼}, incident DoLP

*P*∈[0.2, 1] with a step length of 0.01, and incident AoP

_{in}*α*∈[0°, 180°] with a step length of 1°), the variation of

_{in}*Error*with the

_{max}*PRNU*coefficient is shown in Fig. 11. It is evident that

*Error*of the AoP increases with the increase in

_{max}*PRNU*, while that of the Stokes vector and the DoLP are not sensitive to the change of the

*PRNU*. Further, the

*PRNU*coefficient of typical real-world sensors is less than 5% (typically within 2%); thus, the

*Error*values of the three Stokes parameters affected by the spatial non-uniformity are 0.89%, 1.38%, and 0.99%, respectively. In addition. the

_{max}*Error*of the DoLP and AoP affected by the spatial non-uniformity are 1.37% and 5.34%, respectively. Thus, when calculating the standard deviations of the Stokes vector, the DoLP, and AoP under the influence of spatial non-uniformity, the errors introduced by applying $\overline {\mathbf {\rm M}}$ are at least two orders of magnitude smaller than the ground truth, which shows that the proposed selection rule of the coefficient matrix can result in a good approximation of the ground truth when calculating the estimation error of DoFP polarimeters caused by spatial non-uniformity. The high consistency between the modeled and measured results presented in Section 4.2.2 further verifies the rationality of the selection rule of the coefficient matrix in this study.

_{max}## 5. Conclusion

Sensor noise primarily limits the measurement precision of DoFP polarimeters and has a significant impact on the reconstruction precision of the DoLP, AoP, and other polarization parameters. Thus, this study proposed an error model for DoFP polarimeters based on temporal noise and spatial non-uniformity and the closed-form expressions of the estimation errors of the primary polarization parameters (Stokes vector, DoLP, and AoP) were derived. Compared with the existing models that analyze only the temporal noise of DoFP polarimeters, we modified the normalization condition in traditional calibration methods of DoFP polarimeters and clarified the selection rule of the coefficient matrix in the proposed model. Thus, the errors caused by temporal noise and spatial non-uniformity of DoFP polarimeters can be estimated independently and accurately by applying this proposed model. The existing models that analyze only the temporal noise can also be regarded as a special case of the proposed model under the assumption of an ideal sensor (that is, the sensor is free from spatial non-uniformity). Further, by applying the proposed model, we simulated and analyzed the influence of the camera device parameters and the intensity and polarization statements of the incident light on the measurement precision of DoFP polarimeters. Moreover, the effectiveness of the proposed model was proven by experiments using linearly polarized light on a real-world DoFP polarimeter. The results of this study can be used to guide the selection of device parameters in the design process, precision estimation, and parameter adjustment in actual measurements for DoFP polarimeters.

The primary difference between the division of time (DoT) and DoFP polarimeters is that the former places a single-direction large-size analyzer directly in front of the imaging sensor (the coefficient matrix corresponding to each pixel on the FPA is basically the same), while the latter uses aluminum nanowires directly etched on pixels as analyzers (which is limited by nanomanufacturing techniques, indicating that the coefficient matrix corresponding to each super-pixel is different). Therefore, there is no essential difference between these polarimeters in the error analysis. Thus, in principle, the proposed error model in this study can also be extended to DoT polarimeters. This work has many interesting perspectives, one of which is the generalization of the proposed model for full Stokes vector polarimeters whose polarizing optical elements involve not only linear polarizers, but also Wollaston prisms and retarders. [14,36,37].

## Funding

National Natural Science Foundation of China (61575023).

## Disclosures

The authors declare no conflicts of interest.

## Data Availability

No data were generated or analyzed in the presented research.

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