Polarimetric precision of micropolarizer grid-based camera in the presence of additive and Poisson shot noise

Stéphane Roussel; Matthieu Boffety; François Goudail

doi:10.1364/OE.26.029968

1. Introduction

Polarimetric camera based on micropolarizer grids placed directly in front of the image sensor can implement a division of focal plane (DoFP) polarimeter. Each pixel of the sensor has a different polarimetric sensitivity linked to the element of the micropolarizer array placed just in front of it. If grouped together, neighbor pixels can sense the whole polarization state of the incoming light. In most existing devices, the array is composed of linear polarizers, so that only the linear characteristics of the polarization states (linear Stokes vector) can be measured [1]. Although some micropolarizer arrays featuring circular polarizer have been demonstrated [2–4], we will focus on linear micropolarizer arrays in this paper.

In order to use this type of cameras, the first step is to perform calibration. This operation is more complex than with a standard camera since the polarimetric characteristics of the micropolarizer in front of each pixel have to be calibrated. This issue has been addressed and is well documented [5–7]. However, the camera is a tool for polarimetric measurement, and it is thus important to specify the link between the calibrated characteristics of the micropolarizer array and the obtained estimation precision of the polarimetric parameters in the presence of measurement noise. The purpose of the present article is to address this issue. First, we describe a calibration approach that makes it possible to represent the effect of Poisson shot noise on estimation of polarimetric parameters. Then, we derive the theoretical expressions of the estimation variance of the Stokes vector, the angle of linear polarization and the degree of linear polarization in the presence of both additive noise and Poisson shot noise, as a function of the calibrated micropolarizer characteristics. These values are compared with the variances that would be obtained with ideal micropolarizers in order to quantitatively assess the effect of manufacturing defects on polarimetric performance. These theoretical results are validated by experimental measurements performed with a real-world micropolarizer grid-camera (PolarCam by 4D Technology). Finally, we illustrate how the equations derived in this paper can be used to guide decisions on manufacturing tradeoffs during practical design of micropolarizer grid-based cameras.

2. Calibration of the camera

In this section, we develop a calibration approach for a microplarizer grid-based camera that accounts for both its photometric and polarimetric defects. This approach is based on recent works about calibration of such cameras [5–8]. Its specific purpose is to make it possible to use the calibrated parameters in order to represent the effect of Poisson shot noise on the estimation precision of the “final product” delivered by the camera, namely, the Stokes vector, the angle of polarization and the degree of polarization of the incoming light.

2.1. Calibration

Let us first consider a pixel of the sensor. It is placed in front of a micropolarizer defined by the following three parameters: its angle ϕ, its intensity transmissions in the parallel state t_‖ and in the orthogonal state t_⊥. Let us assume that it is illuminated with light whose polarization state is defined by a linear Stokes vector I₀S, where S^T = (1, S₁, S₂) is a unit intensity Stokes vector $(S_{1}^{2} + S_{2}^{2} = 1)$ and I₀ is the illumination impinging on the pixel, expressed in number of photons. It will produce a digital signal equal to:

d = g η t I_{0} v^{T} S + b

where d is expressed in terms of number of digital levels, b an offset also expressed as a number of digital levels, η is the quantum efficiency of the pixel, g is a detector gain expressed in number of digital levels per photo-electron,

t = t_{‖} + t_{⊥}

is the total transmission of the micropolarizer, and v is its “normalized” analysis vector defined as

v^{T} = \frac{1}{2} [1, q cos (2 ϕ), q sin (2 ϕ)]

where ϕ denotes the angle of the polarizers and

q = \frac{t_{‖} - t_{⊥}}{t_{‖} + t_{⊥}}

the diattenuation, that indicates the “ideality” of the micropolarizer. This coefficient varies between 0 and 1, and is equal to 1 if the polarizer is ideal. The same information can also be represented by the extinction ratio defined as follows:

ζ = \frac{t_{‖}}{t_{⊥}} = \frac{1 + q}{1 - q}

In practice, the parameters of each micropolarizer are spatially varying. Since the pixels are grouped by sets of 4 to form a superpixel (Fig. 1), we will index a pixel as nm, where n ∈ [1, N] denotes the index of the super pixel and m ∈ [1, 4] denotes each of the 4 pixels inside a superpixel. If the incident Stokes vector is spatially uniform and equal to I₀S, the digital signal measured by the camera pixel of index nm is:

d_{n m} = g I_{0} η_{n m} t_{n m} v_{n m}^{T} S + b_{n m}

where η_nm, t_nm, v_nm, and b_nm are respectively the quantum efficiency, the transmission, the normalized analysis vector, and the digital offset of the pixel nm.

Fig. 1 Schematic representation of the micropolarizers on the camera sensor. A superpixel is composed of 4 pixels with micropolarizers oriented at 0°, 45°, 90° and 135°.

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In order to be able to retrieve the incident Stokes vector S from the measurements d_nm, one has to calibrate the camera. This is done by illuminating the camera with spatially uniform light from an integrating sphere and a number K of different input Stokes vectors of expressions A_k, k ∈ [1, K] (the vectors A_k have unit intensities) and intensity I₀. Let us define the K × 3 matrix A in the following way:

A = {[\begin{matrix} A_{1} & A_{2} & \dots & A_{K} \end{matrix}]}^{T}

The K dimensional vector d_nm formed by the obtained measurements at each pixel nm is equal to:

d_{n m} = (g I_{0} η_{n m} t_{n m}) A v_{n m} + b_{n m} 1

where 1 is a K dimensional vector of ones. By inverting this relation, one obtains:

A^{†} (d_{n m} - b_{n m} 1) = g I_{0} η_{n m} t_{n m} v_{n m}

where the superscript † denotes the pseudo-inverse matrix [9]. The offset b_nm of the pixel can be estimated from an image without illumination. The coefficient g can be estimated classically by using several values of illumination level and plotting the temporal variance of the number of digital numbers (minus the offset) as a function of its mean. When the level of light is sufficiently large, this curve becomes a line whose slope is equal to g. [10]. The knowledge of this parameter - which is a parameter of the sensor, not of the micropolarizers - is important for characterizing the precision of measurements in the presence of Poisson noise, as will be seen in the following.

However, the intensity I₀ of the calibration Stokes vectors is difficult to know unless input light from the integrating sphere is precisely calibrated. From Eq. (9), the transmissions η_nmt_nm of each pixel will thus be known only within a multiplicative coefficient. One can choose to normalize them with the average value over the whole sensor of the first coordinate of the inverted signals defined in Eq. (9), that is:

\bar{{Ad}_{0}} = \frac{1}{4 N} \sum_{n = 1}^{N} \sum_{m = 1}^{4} {[A^{†} (d_{n m} - b_{n m} 1)]}_{0} = \frac{1}{2} g I_{0} \bar{η t}

where

\bar{η t} = \frac{1}{4 N} \sum_{n = 1}^{N} \sum_{m = 1}^{4} η_{n m} t_{n m}

since by definition, [v_nm]₀ = 1/2. As a result of this calibration, the polarimetric behavior of each pixel of the camera will be represented by the following normalized vector:

w_{n m} = \frac{A^{†} (d_{n m} - b_{n m} 1)}{2 \bar{A d_{0}}} = \frac{η_{n m} t_{n m}}{\bar{η t}} v_{n m}

The absolute transmission η_nmt_nm of each pixel is unknown, but their relative values are known. This is enough to estimate the useful polarimetric parameters of the light arriving on each superpixel, as will be shown in the next section.

2.2. Estimation of the Stokes vector

Let us now explain how these results can be used to estimate an input Stokes vector defined as

S = S_{0} [1, P cos (2 α), P sin (2 α)]

where S₀ is its intensity, P its degree of linear polarization (DOLP) and α its azimuth, or angle of polarization (AOP). Using the notation defined in the previous section, the measured data for each pixel can be written as:

d_{n m} = g \bar{η t} w_{n m}^{T} S + b_{n m}

Let us denote d_n = [d_n1, d_n2, d_n3, d_n4]^T the 4-dimensional vector that gathers the measured signal for the superpixel n, b_n = [b_n1, b_n2, b_n3, b_n4]^T the vector that gathers the offsets of the superpixel n, and

W_{n} = g {[w_{n 1}, w_{n 2}, w_{n 3}, w_{n 4}]}^{T}

the measurement matrix of superpixel n. It is a 4 × 3 matrix whose rows are the measurement vectors of each pixels of the superpixel n, multiplied by the coefficient g. Eq. (14) becomes:

d_{n} = \bar{η t} W_{n} S + b_{n}

By inverting the relation, one obtains:

\bar{η t} S = W_{n}^{†} (d_{n} - b_{n})

In other words, what can be estimated from the measurements is the Stokes vector expressed in terms of number of photoelectrons, namely,

S^{pe} = \bar{η t} S

, which is proportional to the actual input Stokes vector expressed in terms of photons. In the following, in order to simplify the notation, we will denote S^pe by S, since this is the value that can be measured. We will also denote the measurement vector with offset subtracted as

{\bar{d}}_{n} = d_{n} - b_{n}

With this notation, Eqs. (16) and (17) become:

{\bar{d}}_{n} = W_{n} S and S = W_{n}^{†} {\bar{d}}_{n}

The knowledge of these vectors makes it possible to estimate the two important parameters of the Stokes vector that are the degree of linear polarization (DOLP)

P = \frac{\sqrt{S_{1}^{2} + S_{2}^{2}}}{S_{0}}

and the angle of polarization (AOP).

α = \frac{1}{2} arctan [\frac{S_{2}}{S_{1}}]

In order to illustrate this approach, we have performed polarimetric calibration of a PolarCam camera manufactured by 4D technology [11, 12]. The measurement matrix W_n of each super pixel has been estimated, and the coefficient g has been found equal to 1/7.85 digital levels per photoelectron. Once the camera is calibrated, one can draw maps of the superpixel parameters, yielding a cartography of the defects of the matrix [6]. For example, we have represented in Fig. 2 the maps of extinction ratio ζ. There are 4 different maps, each of them corresponding to one of the 4 possible orientations of the micro-polarizer in front of the pixel. It is seen that the average extinction ratio is slightly different for each value of the orientation. Moreover, the estimated value of the extinction ratio is better in the center of the sensor than on the edge. This may be due to the fact that there is more crosstalk between the pixels on the side of sensor because the light is more divergent [13]. Another possible reason is the spatial inhomogeneity of the illumination during the measurements. The maps of extinction ratio show some similarities with the map found in [6] where the extinction ratio is strongest in the center of the sensor. The mean values of the extinction ratio in Table 1 are slightly lower than in [5] where the average extinction ratio is about 20 (26dB). The difference could be due to the fact that we did not take into account the extinction ratio of the polarizer we use to calibrate the camera. We have also represented in Fig. 3 the estimated micro-polarizer angle for the 4 possible nominal values of this angle. The orientation of the micro-polarizers also shows dispersion and spatial inhomogeneity on the sensor because of the way the micro-polarizers are fabricated. One also notices in Table 1 a shift of ≈ 1.5° that is common to the 4 orientations and that may be due to a residual angle of the polarizer we used to perform the measurement.

Fig. 2 Maps of the extinction ratio of the micro-polarizers on the sensor. a) 0°, b) 45°, c) 90°, d) 135°.

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Table 1. Average orientations and extinction ratio of the micro-polarizers.

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Fig. 3 Maps of the orientations of the micro-polarizers on the sensor. a) 0°, b) 45°, c) 90°, d) 135°.

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3. Modeling and evaluation of polarization estimation errors due to noise

The analysis of the camera’s defects presented in the previous section is useful to characterize the quality of the micropolarizer manufacturing process. However, for practical operation of an existing camera, the main use of calibration results is to correct the image from manufacturing errors. Thanks to calibration, the measurement bias due to these defects can be corrected, but even corrected, these defects still have an influence on the value of the estimation errors due to noise. Indeed, it is well known that the estimation errors on the linear Stokes vector, the DOLP and the AOP depend on the “conditioning” of the analysis matrix [14–16], and this matrix is different for each superpixel due to manufacturing errors. The purpose of the present section is to model and characterize this dispersion of the estimation errors among the superpixels. Estimation errors due to noise when the 4 analysis vectors are ideal and oriented exactly 45° apart have been studied in [16]. In order to represent the estimation precision of the micropolarizer camera, we have to generalize this study to the case of non-ideal polarizers.

It is to be noted that the estimation precision of Stokes vector parameters with a division of amplitude polarimeter has been addressed in the presence of detection noise and of different bias sources [17,18]. In the present paper, we concentrate on scenarios where detection noise is the dominant source of inaccuracies. This will enable us to derive closed-form expressions of the estimation variances not only of the Stokes vectors, but also of the DOLP and of the AOP, for micropolarizer grid-based division of focal plane polarimeters.

3.1. Estimation of the Stokes parameters

To estimate S from d̄, we will use the pseudo-inverse estimator.

\hat{S} = W^{†} \bar{d}

where W denotes the measurement matrix of a superpixel (in the following, we will drop the subscript n of W_n in order to simplify notation). Since the measurements are perturbed by noise, d̄ is a random vector and so is the estimator Ŝ of the Stokes vector. Digital images are corrupted by two main sources of noise: additive noise and Poisson shot noise, the latter being in most cases the dominant noise source. In the presence of these noise sources, which are statistically independent from each other, d̄ is a random vector such that [19]:

VAR [\bar{d}] = g W S + g^{2} σ_{a}^{2}

where VAR [.] denotes the variance of a random vector, σ_a denotes the standard deviation of the additive noise expressed in number of photoelectrons, and g is the number of digital levels per photoelectron. From the properties of additive and Poisson shot noise, the fluctuations are statistically independent from one intensity measurement to the other and the covariance matrix Γ^d of d̄ is thus diagonal.

It is worth to note that in the presence of Poisson noise, the pseudo-inverse estimator is not equivalent to the maximum likelihood (ML) estimator [20]. However, we will use in this paper the pseudo-inverse estimator since it is a simple closed-form algorithm that gives good results in practice. Moreover, it has been shown in [21] that when the measurement matrix is correctly balanced, the pseudo-inverse estimator leads to estimation variance very close to that of the ML estimator.

It has been shown in [19] that if the only perturbations that affect the measurements are additive and Poisson noise, the estimator defined in Eq. (22) is unbiased. In practical situations, such phenomena as chromatic dispersion and the finite bandwidth of incident light would introduce a bias in the estimation [17]. However, in this article, our purpose is to focus on the effect of detection noise, and we will consider situations where this bias can be neglected. The precision of the estimator defined in Eq. (22) precision can thus be represented by its covariance matrix Γ^Ŝ. The variances of the three components of the linear Stokes vector are given by the diagonal values of this covariance matrix. One can synthetically characterize the estimation performance by the sum of these variances, i.e., the trace of the covariance matrix which is also called the equally weighted variance (EWV):

EWV = trace [Γ^{\hat{S}}]

Since the additive and Poisson noise sources are statistically independent, the covariance matrix can be written as

Γ^{\hat{S}} = Γ^{add} + Γ^{poi}

where Γ^add is the covariance matrix in the presence of additive noise only, and Γ^poi the covariance matrix in the presence of Poisson noise only. These two matrices have the following expressions [15,19]:

Γ_{i j}^{add} = σ_{a}^{2} δ_{i j} and Γ_{i j}^{poi} = \sum_{k = 0}^{2} S_{k} γ_{i j}^{k}

where

δ_{i j} = g^{2} {[{(W^{T} W)}^{- 1}]}_{i j} and γ_{i j}^{k} = g \sum_{l = 1}^{4} W_{i l}^{†} W_{j l}^{†} W_{l k}, \forall (k, i, j) \in {[0, 2]}^{3}

Let us define the ideal configuration of a superpixel as the configuration where the analysis vectors are ideal (i.e with equal transmissions and unit diattenuations) and oriented exactly at angles 0, 45, 90, 135 degrees. In this case, the expressions of the covariance matrices in the presence of additive and Poisson noise sources are [16]:

Γ^{add} = σ_{a}^{2} [\begin{matrix} 1 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 2 \end{matrix}] and Γ^{poi} = \frac{1}{2} [\begin{matrix} S_{0} & S_{1} & S_{2} \\ S_{1} & 2 S_{0} & 0 \\ S_{2} & 0 & 2 S_{0} \end{matrix}]

In this ideal case, it is easily seen that

{EWV}_{ideal} = 5 (σ_{a}^{2} + \frac{S_{0}}{2})

Consequently, in the ideal case, the EWV only depends of the input Stokes vector through its intensity S₀, and not its AOP nor its DOLP. This is no longer the case when the four analyzers are not ideal and modeled as in Section 2. Indeed, in the general case, Eq. (26) leads to:

EWV = σ_{a}^{2} \sum_{i = 0}^{2} δ_{i i} + \sum_{k = 0}^{2} S_{k} β_{k} with β_{k} = \sum_{i = 0}^{2} γ_{i i}^{k}

Using the parametrization of S defined in Eq. (13), this relation can also be written:

EWV = σ_{a}^{2} \sum_{i = 0}^{2} δ_{i i} + S_{0} β_{0} {1 + C cos [2 (α - θ)]}

with

θ = \frac{1}{2} arctan [\frac{β_{2}}{β_{1}}] and C = P \frac{\sqrt{β_{1}^{2} + β_{2}^{2}}}{β_{0}}

It is seen in Eq. (31) that the EWV consists of the sum of two terms. The first one is independent of S₀, and corresponds to the contribution of additive noise. The second one is proportional to S₀, and corresponds to the contribution of Poisson noise. This second term varies sinusoidally with the AOP α, the phase θ of this sinusoid depending on the coefficient β_k, and its contrast C depending on the β_k and of the degree of polarization P of the input Stokes vector. One can note that this periodic variation of the Stokes parameter’s variance has already been pointed out in [18]. One can define an “average” level of EWV, averaged over all AOP values, in the following way:

\bar{EWV} = \frac{1}{π} \int_{0}^{π} EWV (α) d α = σ_{a}^{2} \sum_{i = 0}^{2} δ_{i i} + S_{0} β_{0}

We will now apply these equations to the characterization of the estimation performance of a real polarimetric camera. For that purpose, we consider a superpixel of the camera presented in Section 2. Its calibrated parameters are as follows:

Angles: {1.2, 46.4, 91.0, 136.4} degrees.
Diattenuations (q): {0.89, 0.90, 0.87, 0.85}
Relative transmissions ( $η_{n m} t_{n m} / \bar{η t}$ ): {1.01, 1.04, 1.05, 0.941}

The measurement matrix W of this pixel can be computed from these data, and from this matrix, the parameter θ and C. Moreover, the additive noise level of the camera is σ_a = 16 photoelectrons and g = 1/7.85. We have plotted in Fig. 5(a), as a function of α (blue dotted line), the ratio EWV(α)/EWV_ideal, that is, the ratio of the EWV obtained with the actual superpixel against the EWV_ideal that would be obtained with a superpixel in the ideal configuration. We also plotted the average level $\bar{EWV} / {EWV}_{ideal}$ (red dotted line). We can conclude that the manufacturing imperfections of this superpixel multiply EWV by a factor 1.2. The variation of this ratio around its average value as a function of the input AOP is quite slight, of the order of ±4%.

Fig. 4 a) Average raw intensity captured captured by the camera during the measurement. b) $\bar{EWV} / {EWV}_{ideal}$ . c) $\bar{VAR [α]} / VAR {[α]}_{ideal}$ . d) $\bar{VAR [P]} / VAR {[P]}_{ideal}$ .

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In order to experimentally validate this result, we estimated the EWV when this superpixel is illuminated by a Stokes vector with DOLP P = 1 and variable AOP α by performing the measurement of the Stokes vector 10⁴ times for each value of the AOP, and estimating the EWV from these 10⁴ realizations. We have plotted in Fig. 5(a) the ratio of this experimental variance to EWV_ideal (blue solid line) and the average over all values of α (red solid line). It is verified that the experimental variance accurately follows the sinusoidal variation predicted by the theory. To get a synthetic view of the estimation precision reached by all the superpixels of the camera, we have plotted in Fig. 4(b) the value of the EWV averaged on all values of α, that is, $\bar{EWV} / {EWV}_{ideal}$ , for each superpixel. One observes a dispersion of this ratio, which ranges from 1.2 (this was the case for the superpixel considered in Fig. 5) to 1.8 in the edges of the sensor. It can also be observed that the map of EWV is closely related to the map of relative sensitivity of the pixels represented in Fig. 4(a). Please notice that the sharp discontinuity in sensitivity between the left and right side of the sensor is due to the fact that the sensor uses two different analog digital converters. This discontinuity is not visible in the extinction ratio maps in Fig. 2 nor the angle of the micro-polarizers in Fig. 3 because these values are independent from the gain of the sensor.

Fig. 5 a) EWV divided by the ideal EWV. b) Variance of AOP divided by ideal variance of AOP. c) Variance of the DOLP divided by the ideal variance of the DOLP. Experimental (solid lines) and theoretical (dotted lines) values are represented in blue and their mean values in red.

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3.2. Estimation of the angle of polarization

Let us now consider estimation of the AOP. The expression of this parameter as a function of the Stokes vector is given in Eq. (21). It is a nonlinear function of the Stokes vector and it is thus difficult to obtain a closed-form expression of its variance. However, one can determine an approximate value of this variance that is valid in the case of small perturbations. Indeed, let X be a K-dimensional random vector with mean <X> and covariance matrix Γ^X, and let y = f(X) be a random variable that is a function of this vector. If the variations of X around <X> are sufficiently small and the function f is sufficiently “smooth” around <X>, then [22]:

< y > ≃ f (< X >) and VAR [y] ≃ {[\nabla f (< X >)]}^{T} Γ_{X} \nabla f (< X >)

where ∇f(x) = [∂f/∂X₁(x), . . ., ∂f/∂X_K(x)]^T is the gradient of the function f. The expression of the gradient of Eq. (21) is [16]:

\nabla \hat{α} = \frac{1}{2 P^{2} S_{0}^{2}} {[\begin{matrix} 0, & - S_{2}, & S_{1} \end{matrix}]}^{T}

The approximate variance of α̂ is obtained by substituting Eqs. (35) and (26) into Eq. (34). In the ideal configuration, this results in:

VAR {[\hat{α}]}_{ideal} = \frac{1}{2 P^{2}} (\frac{σ_{a}^{2}}{S_{0}^{2}} + \frac{1}{2 S_{0}})

It it is seen that this variance is independent of the actual value of the AOP. It depends only on σ_a, S₀, and P². This is no longer the case in a non-ideal configuration. Indeed, substituting Eqs. (35) and (26) in Eq. (34), and making use of Eq. (13), one obtains after cumbersome but elementary calculations:

\begin{matrix} VAR [\hat{α}] & = & \frac{σ_{a}^{2}}{4 P^{2} S_{0}^{2}} {δ_{11} s^{2} + δ_{22} c^{2} - 2 δ_{12} c s} + \frac{1}{4 P^{2} S_{0}} {γ_{11}^{0} s^{2} + γ_{22}^{0} c^{2} - 2 γ_{22}^{2} c s + \\ {Pc}^{2} [(γ_{22}^{2} - 2 γ_{12}^{1}) s + γ_{22}^{1} c] + {Ps}^{2} [(γ_{11}^{1} - 2 γ_{12}^{2}) c + γ_{11}^{2} s]} \end{matrix}

where the following notation has been used:

c = cos (2 α) and s = sin (2 α)

It is seen in Eq. (37) that VAR[α̂] is the sum of two terms. The first one is inversely proportional to the ratio (S₀/σ_a)², which can be seen as the intensity signal to noise ratio (SNR) in the presence of additive noise. The second term is inversely proportional to S₀, which is the intensity SNR in the presence of Poisson noise. Thus for low light levels, that is, for low values of S₀, the estimation variance of the AOP varies as

1 / S_{0}^{2}

, whereas for larger light levels, it varies as 1/S₀. It is also seen that VAR[α̂] is a weigthed sum of trigonometric functions of α: the estimation variance thus depends on the actual value of α. The amplitude of this variation with respect to α depends on the degree of polarization P of the incident Stokes vector, on S₀, σ_a and on the measurement matrix W through the coefficients δ_ij and

γ_{i j}^{k}

. One can define an “average” variance of α̂, averaged over the azimuth values, in the following way:

\bar{VAR [\hat{α}]} = \frac{1}{π} \int_{0}^{π} VAR [\hat{α}] (α) d α = \frac{1}{8 P^{2}} [\frac{σ_{a}^{2}}{S_{0}^{2}} (δ_{11} + δ_{22}) + \frac{1}{S_{0}} (γ_{11}^{0} + γ_{22}^{0})]

For illustration, we have considered the same superpixel of the camera as in the previous section. We have plotted in Fig. 5(b), as a function of α for P = 1 (blue dotted line), the ratio VAR[α̂]/VAR[α̂]_ideal, that is, the ratio of the of the AOP estimation variance obtained with the actual camera against the variance that would be obtained with a perfect camera. We also plotted the average level

\bar{VAR [\hat{α}]} / VAR {[\hat{α}]}_{ideal}

(red dotted line). We can conclude that the manufacturing imperfections of this superpixel multiply the AOP estimation variance by a factor 1.3. The variation of this ratio around its average value as a function of the input AOP is of the order of ±10%.

In order to experimentally validate this result, we estimated the AOP estimation variance when this superpixel is illuminated by a Stokes vector with DOLP P = 1 and variable AOP from 10⁴ measurements of the Stokes vector for each value of the AOP. We plotted in Fig. 5(b) the ratio of this experimental variance and VAR[α̂]_ideal (blue solid line), and the average over all values of α (red solid line). It is verified that the experimentally estimated variance follows quite well the variation of the theoretical one. In order to get a synthetic view of the performance of the camera in terms of AOP estimation, we have represented in Fig. 4(c) the ratio VAR [α]/VAR[α]_ideal. We observe that this ratio ranges from 1.3 to 2, and that its spatial distribution is closely related to the sensitivity map of the sensor (Fig. 4(a)).

3.3. Estimation of the degree of linear polarization

Let us now turn to the estimation of the DOLP. The expression of the gradient of Eq. (20) is [16]:

\nabla \hat{P} = \frac{1}{P S_{0}^{2}} {[\begin{matrix} - P^{2} S_{0}, & S_{1}, & S_{2} \end{matrix}]}^{T}

The approximate variance of P̂ is obtained by substituting Eqs. (40) and (26) into Eq. (34). In the ideal case, where the covariance matrices are given by Eq. (28), one obtains [16]:

VAR {[\hat{P}]}_{ideal} = \frac{σ_{a}^{2}}{S_{0}^{2}} [2 + P^{2}] + \frac{1}{2 S_{0}} [2 - P^{2}]

It is seen that this variance is independent of the actual value of the AOP, and only depends on σ_a, S₀ and P². This is no longer the case in a non ideal configuration. Indeed, substituting Eqs. (40) and (26) in Eq. (34), and making use of Eq. (13), one obtains after cumbersome but elementary calculations:

\begin{matrix} VAR [\hat{P}] & = & \frac{σ_{a}^{2}}{S_{0}^{2}} {P^{2} δ_{00} - 2 P (δ_{01} c + δ_{02} s) + 2 δ_{12} c s + δ_{11} c^{2} + δ_{22} s^{2}} + \\ \frac{1}{S_{0}} {P^{3} (γ_{00}^{1} c + γ_{00}^{2} s) + P^{2} [γ_{00}^{0} - 2 γ_{01}^{1} c^{2} - 2 γ_{02}^{2} s^{2} - 2 (γ_{01}^{2} + γ_{02}^{1}) c s] + \\ P [(γ_{11}^{2} + 2 γ_{12}^{1}) c^{2} s + (γ_{22}^{1} + 2 γ_{12}^{2}) c s^{2} + γ_{11}^{1} c^{3} + γ_{22}^{2} s^{3} - 2 γ_{01}^{0} c - 2 γ_{02}^{0} s] + \\ [γ_{11}^{0} c^{2} + γ_{22}^{0} s^{2} + 2 γ_{12}^{0} c s]} \end{matrix}

It is seen that similarly to VAR[α̂], VAR[P̂] is the sum of a term inversely proportional to (S₀/σ_a)², that corresponds to the additive noise contribution, and a second one proportional to S₀, that corresponds to the Poisson noise contribution. Both terms are weigthed sums of trigonometric functions of α: the estimation variance thus depends on the actual value of α. The amplitude of this variation with respect to α depends on the degree of polarization P of the incident Stokes vector, on S₀, σ_a and on the measurement matrix W through the coefficients δ_ij and

γ_{i j}^{k}

. where the notation defined in Eq. (38) has been used. One can define an “average” variance of P̂ over the azimuth values in the following way:

\begin{array}{l} \bar{VAR [\hat{P}]} & = & \frac{1}{π} \int_{0}^{π} VAR [\hat{P}] (α) d α \\ = & \frac{σ_{a}^{2}}{S_{0}^{2}} [P^{2} δ_{00} + \frac{δ_{11} + δ_{22}}{2}] + \frac{1}{2 S_{0}} [(γ_{11}^{0} + γ_{22}^{0}) + 2 P^{2} (γ_{00}^{0} - γ_{01}^{1} - γ_{02}^{2})] \end{array}

For illustration, we have considered the same superpixel of the camera as previously. We have plotted in Fig. 5(c), as a function of α (blue dotted line), the ratio VAR[P̂]/VAR[P̂]_ideal, that is, the ratio of the of the DOLP estimation variance obtained with the actual camera against the variance that would be obtained with a perfect camera. We also plotted the average level

\bar{VAR [\hat{P}]} / VAR {[\hat{P}]}_{ideal}

(red dotted line). We can conclude that the manufacturing imperfections of this superpixel multiply the DOLP estimation variance by a factor 1.6. The variation of this ratio around its average value as a function of the input AOP is of the order of ±25%, thus significantly larger than the variation of AOP estimation variance.

In order to experimentally validate this result, we estimated the DOLP estimation variance when this superpixel is illuminated by a Stokes vector with DOLP P = 1 and variable AOP from 10⁴ measurements of the Stokes vector for each value of the AOP. We plotted in Fig. 5(c) the ratio of this experimental variance to VAR[P̂]_ideal (blue solid line) and the average over all values of α (red solid line). It is verified that the experimentally estimated variance follows quite well the variation of the theoretical one. We have represented in Fig. 4(d) the ratio VAR[P]/VAR[P]_ideal for all the superpixels of the camera. We observe that its value ranges from 1.6 to 3, and its spatial distribution is closely related to the sensitivity map of the camera, the worst precision being reached at the edges of the sensor.

To summarize the results obtained in this section, we have shown that it is possible to predict the polarimetric estimation performance of each superpixel of a micropolarizer grid-based camera in the presence of additive and Poisson shot noise by using the results of its calibration. It is noticed that manufacturing imperfections can lead to an increase of the estimation variance. It is also apparent that with a non-ideal superpixel, the estimation variance does depend on the AOP of the input state. This variation can be experimentally observed, although it is slight compared to the average value of the variance.

4. Discussion

The main results obtained in this article are contained in Eqs. (37) and (42). These equations reveal the influence of the different sources of manufacturing uncertainties on the final product delivered by the sensor, that is, the estimated AOP and DOP. In this section, we illustrate how these equations can be used to analyze and compare the influence of the different sources of uncertainty on the estimation precision, and thus help engineering choices when designing a micropolarizer grid-based camera.

A first possible use of these equations is to assess the influence of light level and read noise on estimation precision. We have plotted in Fig. 6(a) the variation of the estimation variance of the AOP averaged on the azimuth values, $\bar{VAR [\hat{α}]}$ , as a function of the light level S₀, for the super pixel parameters considered as an example in Section 3, and for different values of the additive noise standard deviation σ_a. The graph is plotted in log-log scale so that we can clearly see a classical trend for polarimetric imaging with digital sensors [23]. At low light levels, the estimation variance decreases quadratically with light level since additive noise is dominant, so that in log-log plot, it is a line of slope −2. For large light levels, it decreases linearly since photon noise is dominant, so that in log-log plot, it is a line of slope −1. The “turnover” between the two regimes occurs when the two SNRs appearing in Eq. (37) are equal. It is verified in Fig. 6(a) that the position of this turnover depends on the value of σ_a.

Fig. 6 a) $\bar{VAR [\hat{α}]}$ as a function of S₀ for different values of σ_a. b) VAR[α̂]_max/VAR[α̂]_ideal as a function of micropixel angle fluctuation amplitude Δ_ϕ.

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The obtained equations can also be used to evaluate the consequences of manufacturing uncertainties. As an example, we have considered that the angle ϕ of each of the four pixels inside a superpixel can vary from their nominal orientation {0°, 45°, 90°,135°} because of manufacturing errors. The diattenuations of the pixels are assumed identical and equal to a given value d, and the transmissions are also assumed identical. We further assume that the angle fluctuations of all pixels are statistically independent and uniformly distributed in the interval [−Δ_ϕ/2, Δ_ϕ/2]. Each fluctuation realization of the four pixels leads to a different superpixel configuration. We have considered K = 10⁴ such superpixel configurations, each of which being indexed by the index k ∈ [1, K]. To each of these configurations corresponds an AOP variance VAR[α̂]_k computed from Eq. (37), that also depends on the true value α of the AOP. We then consider:

VAR {[\hat{α}]}_{\max} = max_{k} [max_{α} (VAR {[\hat{α}]}_{k})] .

This represents the variance obtained in worst case, that is, with the worst superpixel configuration and the worst value of the input light AOP. In Fig. 6(b), we have represented VAR[α̂]_max/VAR[α̂]_ideal as a function of Δ_ϕ, for diattenuation values d equal to 1, 0.9, and 0.8. It is seen that VAR[α̂]_max increases as Δ_ϕ increases and and as d decreases, which was expected. The main interest of this graph is to enable one to compare the relative influence of angle fluctuation and diattenuation value on estimation precision. For example, it can be seen that a configuration with diattenuation d = 0.8 and no angle fluctuation (Δ_ϕ = 0) is equivalent to a configuration with larger diattenuation d = 0.9 and angle fluctuation amplitude Δ_ϕ around 7°. In other words, increasing the diattenuation of the pixels makes it possible to relax constraints on angle orientation precision. This type of information is useful to guide decisions on manufacturing tradeoffs.

Of course, the results presented in this section are only given for purpose of illustration. In practice, more precise simulations and analyses can be done with the equations derived in this paper when precise information on the manufacturing conditions and constraints linked to a practical application are known.

5. Conclusion

We have characterized the polarimetric estimation performance of micropolarizer grid-based camera in the presence of additive and Poisson noise. The estimation variances of AOP and DOLP depend on the actual characteristics of each superpixel, which are obtained by calibration, and also on the actual values of the AOP and the DOLP of the input Stokes vector. We derived closed-form expressions of these variances as a function of these parameters. These expressions have been validated by experimental measurement on a commercial polarimetric camera. The obtained values of the variance have been compared with the variances that would be obtained with ideal superpixels in order to quantitatively assess the influence of manufacturing and optical defects on polarimetric performance. With the camera evaluated here, the increase of variance is around 1.5 times, which is relevant if precision measurements have to be performed. We have also illustrated how the equations derived in this paper can be used to analyze and compare the influence of the different sources of uncertainty on estimation precision, and thus help engineering choices when designing a micropolarizer grid-based camera. This work has many interesting perspectives, one of them is to generalize this approach to DoFP cameras that are able to measure the full Stokes vector since their grid involves not only linear polarizers, but also retarders [2–4].

Funding

Agence Nationale de la Recherche (ANR); Direction Générale de l’Armement (DGA) (ANR-16-ASMA-0007-01 POLNOR); Mission pour la Recherche et l’Innovation Scientifique (MRIS).

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Orientation	Extinction ratio

1.45 ± 0.35	12.4 ± 1.78
46.5 ± 0.28	13.5 ± 1.93
91.2 ± 0.46	10.7 ± 1.47
136.5 ± 0.45	9.4 ± 1.31

Polarimetric precision of micropolarizer grid-based camera in the presence of additive and Poisson shot noise

Abstract

1. Introduction

2. Calibration of the camera

2.1. Calibration

2.2. Estimation of the Stokes vector

3. Modeling and evaluation of polarization estimation errors due to noise

3.1. Estimation of the Stokes parameters

3.2. Estimation of the angle of polarization

3.3. Estimation of the degree of linear polarization

4. Discussion

5. Conclusion

Funding

References

Cited By

Figures (6)

Tables (1)

Equations (44)

Optics Express