Handy method to calibrate division-of-amplitude polarimeters for the first three Stokes parameters

Olivier Morel; Ralph Seulin; David Fofi

doi:10.1364/OE.24.013634

1. Introduction

Polarization imaging is an active research area which concerns various fields such as machine vision, medical imaging and defense applications. Initially dedicated for static applications, such as inspection [1–3] and three-dimensional reconstruction of specular objects [4–6], polarization imaging becomes increasingly used to dynamic scenes such as rocket plumes measurement [7] or marine biology studies [8]. With the introduction of real-time polarimetric cameras new applications in the field of mobile robotics can be addressed such as automatic mud detection for unmanned ground vehicles [9]. Three main technologies are used to perform real-time acquisition: active liquid crystals [10], pixelated filter [11] and three-CCD sensors. These groups also refer to the classification into Division-of-time (DoT), Division-of-focal plane (DoFP) and Division-of-amplitude (DoA) polarimeters respectively [12–14]. The last two technologies are more robust in outdoor conditions and are more suitable for embedded applications since all the required components needed to reconstruct the incident polarization state are acquired simultaneously. Contrary to DoA polarimeters, DoFP polarimeters split the image by placing a pixelated filter in front of the sensor and provide polarization images with higher acquisition rate but with lower resolution. The main counterpart of DoA polarimeters is that the three-CCD cameras require precise alignment of the three sensors and this task can only be performed by specialized manufacturers. For both technologies (DoA and DoFP), specific calibrations methods were developed by [15] and [16] respectively. The proposed methods require specific optical devices such as calibrated photodiode, integrating sphere and are quite difficult to implement for non-specialist users. This paper intends to provide a handy method to calibrate efficiency the camera. New areas of application open by the polarimetric cameras often need measurement of the angle of polarization and a relative estimation of the degree of polarization. It is demonstrated that these parameters can be properly estimated using the proposed method with a simple tablet that rotates in front of the camera.

In section 2 the model used to describe the characteristics of the three polarizers is described. Then, the section 3 introduces the proposed method that enables to calibrate the three-CCD polarimetric camera using a linearly polarized source with no prior regarding its intensity and its degree of polarization. Experimental results presented in section 4 show the efficiency of the method for calibrating the camera. Finally, since some tablets provide partially linearly polarized they can be used as a light source to calibrate the camera. An automatic and effective calibration process based on these properties is described in section 5.

2. Three-CCD polarimetric camera model

Generally, the three-CCD polarimetric camera is made of three polarizers that are far from being perfect and that are not necessarily precisely oriented. In the literature, the three orientations for θ_i {0°,45°,90°} are frequently chosen even if it had been demonstrated that the combination {0°,60°,120°} gives better results by improving signal-to-noise ratio [17]. In a general approach, each of the three polarizers can be represented by the Mueller matrix M_i(θ_i) of a real polarizer oriented at the angle θ_i. The Mueller matrix can be written in the following form [18]:

M_{i} (θ_{i}) = T [\begin{matrix} 1 + u + v & \cos 2 θ_{i} & \sin 2 θ_{i} & 0 \\ \cos 2 θ_{i} & (1 + u + v) \cos^{2} 2 θ_{i} & (1 + u + v) \cos 2 θ_{i} \sin 2 θ_{i} & 0 \\ \sin 2 θ_{i} & (1 + u + v) \cos 2 θ \sin 2 θ_{i} & (1 + u + v) \sin^{2} 2 θ_{i} & 0 \\ 0 & 0 & 0 & 0 \end{matrix}],

where T is the average transmittance for the polarization part of transmission (referred as transmittance here), u and v are, respectively, the ratios of the copolarized and cross-polarized incoherent NSS (Near-Specular-Scattering) to T. For an ideal polarizer, u = v = 0, T = 1/2 and M(θ) from Eq. (1) reduces to the form of the Mueller matrix of a perfect polarizer. The degree of polarization of the polarizer can be defined by:

P = 1 / (1 + u + v) .

The polarimetric camera must be able to estimate the three first Stokes parameters of a partially linearly polarized light described by the Stokes vector:

[\begin{matrix} s_{0} \\ s_{1} \\ s_{2} \\ 0 \end{matrix}] = [\begin{matrix} I_{0} \\ I_{0} ρ \cos 2 α \\ I_{0} ρ \sin 2 α \\ 0 \end{matrix}],

where I₀, ρ, α are respectively the light intensity, the degree of polarization and the angle of polarization of the incoming light.

Since the sensors are only sensitive to light intensity, the intensities measured by the camera are given by keeping the first component resulting from the matricial product: M_i · s. A set of three equations is therefore obtained:

\forall i \in [1, 3] P_{i} = T_{i} (\frac{S_{0}}{P_{i}} + s_{1} \cos 2 θ_{i} + s_{2} \sin 2 θ_{i}),

where

P_{i}

, T_i and θ_i are respectively the degree of polarization, the transmittance, and the orientation of polarization for the three polarizers.

The aim of the polarization calibration step here is to estimate the parameters of the three polarizers in order to improve the measurement of the polarization parameters of the incoming light. It is not intended to estimate the full Mueller matrices of the system. Therefore, the three-CCD polarization calibration results in providing an estimation of the angle of the three polarizers θ_i and their degree of polarization and transmittance $P_{i}$ , T_i. Equation (4) can be handled as a linear least squares problem. The three equations can be written in matrix form as:

[\begin{matrix} P_{1} \\ P_{2} \\ P_{3} \end{matrix}] = A \cdot [\begin{matrix} s_{0} \\ s_{1} \\ s_{2} \end{matrix}],

where:

A = [\begin{matrix} \frac{T_{1}}{P_{1}} & T_{1} \cos 2 θ_{1} & T_{1} \sin 2 θ_{1} \\ \frac{T_{2}}{P_{2}} & T_{2} \cos 2 θ_{2} & T_{2} \sin 2 θ_{2} \\ \frac{T_{3}}{P_{3}} & T_{3} \cos 2 θ_{3} & T_{3} \sin 2 θ_{3} \end{matrix}] .

Estimation of the three first Stokes parameters in the least square sense is equivalent of computing:

{(A^{t} A)}^{- 1} A^{t} \cdot [\begin{matrix} P_{1} \\ P_{2} \\ P_{3} \end{matrix}],

Consequently, the polarizers properties can be soundly encoded in a polarization calibration matrix H defined as follows:

H = {(A^{t} A)}^{- 1} A^{t} .

In a real-time process, this polarization calibration matrix H is computed offline and the three first Stokes parameters images can be directly computed using Eq. (7).

3. Calibration using a polarized light source

To create ground truth and describe principle of the proposed method, a linear polarizer (called analyzer here) placed in front of the camera is used to produce linearly polarized light with a known orientation α as presented in Fig. 1. Considering also that this analyzer is not necessarily perfect, the produced light will be partially linearly polarized and can be described by the Stokes vector introduced in Eq. (3). The light measured by a corresponding pixel on the three sensors is given by:

\begin{array}{l} \forall i \in [1, 3] P_{i} = \frac{T_{i}}{P_{i}} I_{0} + T_{i} I_{0} ρ \cos 2 θ_{i} \cos 2 α + T_{i} I_{0} ρ \sin 2 θ_{i} \sin 2 α \\ = a_{i} + b_{i} \cos 2 α + c_{i} \sin 2 α . \end{array}

Fig. 1 Calibration set-up. An analyzer is placed between the polarization camera and a uniform light source. The polarization camera is made of three linear polarizers (represented by red, green and blue colors) placed with different orientation in front of the three sensors.

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If the analyzer takes n different orientations α⁽¹⁾, α⁽²⁾,…,α⁽ⁿ⁾, a set of n equations will be obtained for each sensor:

{\begin{array}{l} P_{i}^{(1)} & = a_{i} + b_{i} \cos 2 α^{(1)} + c_{i} \sin 2 α^{(1)} \\ P_{i}^{(2)} & = a_{i} + b_{i} \cos 2 α^{(2)} + c_{i} \sin 2 α^{(2)} \\ ⋮ & ⋮ \\ P_{i}^{(n)} & = a_{i} + b_{i} \cos 2 α^{(n)} + c_{i} \sin 2 α^{(n)} \end{array}

For each sensor, the parameters a_i, b_i and c_i can be estimated using a least mean square approach. Consequently, the θ_i angles of the polarizers can be directly computed by using:

θ_{i} = \frac{1}{2} {arctan}_{2} (c_{i}, b_{i}) .

As a result, it is important to notice that even if the analyzer used for the calibration process is not a perfect linear polarizer, the angle of the three polarizers can be suitably estimated. Since the quantities I₀ and ρ are supposed to be unknown it is impossible to estimate all the parameters T_i and

P_{i}

for the three sensors. Nevertheless, the degree of polarization of each sensor can be expressed as:

ρ P_{i} = \frac{\sqrt{b_{i}^{2} + c_{i}^{2}}}{a_{i}} .

As a consequence, the degree of polarization estimation of each sensor is directly related to the degree of polarization of the analyzer used for calibration. Similarly we have:

T_{i} = \frac{i}{I_{0}} a_{i} P_{i},

leading to the conclusion that the average transmittance for the polarization part of transmission cannot be estimated for the three sensors. The parameters T_i and

P_{i}

can be considered relatively against each other. As it will be shown later, the knowledge of their relative values significantly improves the polarization parameters estimation. Without loss of generality, the following approaches can be considered:

The first polarizer of the camera is perfect $(P_{1} = 1)$ and (T₁ = 0.5) and the other polarizer parameters are computed relatively to this one. In this case, the degree of polarization and the transmittance values can exceed their physical values 1 and 0.5 respectively. From the estimated values, the parameters are derived according to: ${\begin{array}{l} P_{1} = 1, & P_{2} = \frac{a_{1}}{a_{2}} \sqrt{\frac{b_{2}^{2} + c_{2}^{2}}{b_{1}^{2} + c_{1}^{2}}}, P_{3} = \frac{a_{1}}{a_{3}} \sqrt{\frac{b_{3}^{2} + c_{3}^{2}}{b_{1}^{2} + c_{1}^{2}}} \\ T_{1} = \frac{1}{2}, & T_{2} = \frac{a_{2} P_{2}}{2 a_{1} P_{1}}, T_{3} = \frac{a_{3} P_{3}}{2 a_{1} P_{1}} \end{array},$
One of the three polarizers of the camera is perfect in terms of degree of polarization $(\max P_{i} = 1)$ and one of the three polarizers is perfect in terms of transmittance $(\max T_{i} = \frac{1}{2})$ . In some cases, one of the three polarizers can have both properties.
The degree of polarization of the analyzer used for calibration is perfect (ρ = 1) and one of the three polarizers is perfect in terms of transmittance $(\max T_{i} = \frac{1}{2})$ .

In the presented cases, it is obvious that the estimated parameters have no strong physical meaning but at least it enables to take into consideration the relative relationship between the polarization parameters of the three polarizers. Considering the light source as a perfect linear light will simply underestimate the degree of polarization of each sensor. As a result, the degree of polarization of the scene maybe overestimated leading sometimes to get a degree of polarization greater than 1. In most application of polarimetric vision in robotics, the degree of polarization is considered as a relative parameter [19].

4. Experimental results

The proposed framework for calibrating a three-CCD polarization camera is tested using a near-perfect linear polarizer as an analyzer placed between the camera and a uniform and diffuse light source as presented Fig. 1. The analyzer is precisely rotated with 10° steps from −90° to 90° according to the vertical plane. From the images obtained onto the three CCD sensors, the average intensity is computed and the sinusoidal relationship between the analyzer angle and the light measured by the sensors can be depicted as presented Fig. 2. As presented in section 3, parameters of the three sinusoids are computed using the least mean square method. The sinusoid approximations for every corresponding pixels were also performed and are represented by the error bars. It highlights that no radiometric calibration was performed here to minimize the spatial non-uniformity of the three sensors. Nevertheless, each of the three po-larizers are assumed to be homogeneous and the method focuses only on the estimation of the polarizers properties.

Fig. 2 Approximation of the sinusoid parameters during the calibration process. Error bars are at ±1 standard deviation and err represents the approximation error.

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From the parameters of the sinusoid, the camera is calibrated according to the proposed framework. Then from the set of three images, the polarization parameters of the incoming light source can be estimated in turn by the polarimetric camera. If the camera is properly calibrated, we should be able to find original angles of the analyzer and since the incident light source and the analyzer remain the same during the experiment a constant value of light intensity and degree of polarization are expected. Five scenarios were considered here to compute the polarization parameters of the incident light source:

No calibration: the 3 polarizers in front of the sensors are assumed to be perfect and their orientations are known (provided by the data-sheet of the camera)
The estimation of the angle of the 3 linear polarizers is used and nevertheless we assume that both the polarizers and sensors responses are perfect
Our calibration method is used according to approach (1)
Our calibration method is used according to approach (2)
Our calibration method is used according to approach (3)

After the calibration process, the camera is supposed to be able to measure polarization parameters and especially those from the incident light source provided by the analyzer. Under these conditions, the five calibration scenarios are tested on data extracted from the three pixels of a same position on the CCDs. It is expected that all Angle of Polarization (AoP) values are equal to the analyzer orientations and that all Degree of Polarization (DoP) are constant.

Figure 3 represents the difference between the orientations of the analyzer and the Angle of Polarization values estimated by the three corresponding pixels of a same position. It appears clearly that knowing the orientation of the polarizers in front of the three CCDs is not enough and leads to an error as important as if no calibration were performed. Using our method with one of the three introduced approaches leads to an error of less than 0.2° which is pretty good.

Fig. 3 Angle difference between the analyzer orientation and the measured Angle of Polarization (AoP) by the three corresponding pixels of a same position.

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Since the light source power was constant and the analyzer was the same during the experiments, it is expected that the Degree of Polarization (DoP) values measured by the three corresponding pixels of a same position presented Fig. 4 remain constant for every orientation of the analyzer. It also shows that the knowledge about orientations of the three polarizers is not enough and it highlights that the proposed polarization calibration method proves essential to accurately measure the polarization parameters with a three-CCD polarimetric camera. Depending on the used approach (1), (2) or (3) the results are different since the degree of polarization and the transmittance of each polarizer in front of the three sensors are computed relatively against each other.

Fig. 4 Degree of Polarization (DoP) of the linearly polarized light source used to calibrate the camera measured by the three corresponding pixels of a same position.

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Table 1 presents the statistical analysis of the polarization parameters estimation. Columns highlighted in gray emphasize the parameters to be considered for comparing the different methods. Indeed the mean of the difference between AoP and analyzer orientation must be as small as possible whereas the standard deviation of DoP and Intensity must be taken into consideration and be close to zero. Since the same lighting source is used here for experimentation, DoP and Intensity must be constant whatever the orientation of the analyzer.

Table 1. Statistical analysis of the polarization parameters estimation of the incident lighting source using the three corresponding pixels of a same position.

View Table

5. Calibration method using a tablet

5.1. Methodology

As demonstrated in the previous section, a partially linearly polarized light source is sufficient to perform the calibration of the polarimetric camera. Therefore, using a tablet that provides partially polarized light such as an Apple iPad instead of a rotating polarizer associated to a lighting source would be a convenient solution. To implement this original solution, the orientation of the iPad that has to be known is computed using the geometric camera calibration process. Consequently, using an iPad displaying a calibration pattern such as the well known chessboard is twofold advantage. On the one hand, the geometric calibration of the camera that is required for 3D reconstruction purpose is directly performed and on the other hand, the orientation of the iPad which is related to the angle of polarization can be estimated for the polarization calibration. The process can be decomposed into 5 main steps as described in Fig. 5.

Fig. 5 Geometric and polarimetric calibration of a polarimetric camera using an iPad.

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The first step is to take images with different positions and orientations of the iPad with respect to the optical axis of the camera. Figure 6 shows images acquired from the three sensors. This step provides k triplets of images (Im₁, Im₂, Im₃) obtained from the three sensors. k must be large enough to perform the geometric calibration of the camera and the orientations have to vary adequately to obtain accurate polarization calibration. Before applying the polarization calibration process that is broadly described in section 2, three main tasks must be performed. They are described in the following subsections.

Fig. 6 Sequences of images acquired on the three-CCD polarimetric camera.

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5.1.1. Automatic corners detection

Since the tablet provides polarized light, to ensure a perfect detection of the corners, an image of intensity that is roughly independent of the orientation of the iPad must be computed. Assuming that the camera have polarizers oriented approximately according to 0°, 60° and 120°, the resulting image $\tilde{I} m$ can be given by:

\tilde{I} m = \frac{I m_{1} + I m_{2} + I m_{3}}{3} .

In the case of unknown orientation of the polarizers, the following equation can also be logically suitable for corners detection:

\tilde{I} m = \max (I m_{1}, I m_{2}, I m_{3}) .

Therefore, from the k triplets of images (Im₁, Im₂, Im₃) k average images are computed using Eq. (15) or Eq. (16) and are provided as inputs for the automatic corner detection algorithm. The output will be a set of m corners coordinates with m ≤ k since in some images the pattern may not be properly detected.

5.1.2. Geometric calibration

From the set of corners, the classical geometric calibration algorithm [20, 21] is performed to estimate the camera matrix K that contains the intrinsic parameters of the camera. It also provides a set of extrinsic parameters (t_vec, r_vec) that will be useful to estimate the rotation of the chessboard relatively to the camera.

5.1.3. Extraction of relevant data

The purpose of this step is to provide both the set of n different angles ${[α^{(j)}]}_{j \in [1, n]}$ and the set of n intensity values from the three sensors ${[P_{1}^{(j)}; P_{2}^{(j)}; P_{3}^{(j)}]}_{j \in [1, n]}$ that are required for the polarization calibration stage.

Once the camera is geometrically calibrated, the orientations of the iPad ${[α^{(j)}]}_{j \in [1, k]}$ can be easily determined using extrinsic parameters. In addition, the angle between the iPad normal and the optical axis of the camera, namely tilt, is computed to keep only values for which the tablet is parallel to the camera image plane to ensure an accurate measurement of the polarized light source.

To have the best signal noise ratio, it is important to provide light with the maximum of intensity. Consequently, the polarization calibration step is not performed on the whole image because of the dark area produced by the black squares. From the detected corners, the centers of the white squares (presented as yellow dots in Fig. 7) are computed and using an averaging neighborhood around the centers an intensity value $P_{i}^{(j)}$ is computed for each images Im_i with i ∈ [1,3]. This step will provide the required values ${[P_{1}^{(j)}; P_{2}^{(j)}; P_{3}^{(j)}]}_{j \in [1, n]}$ for the polarization calibration process.

Fig. 7 Orientation estimation and average intensities extraction.

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5.2. Experiments

5.2.1. Tablet properties

During experiments, an Apple iPad 2 was used as a polarized lighting source. To ensure that the tablet provides polarized light, a photodiode and a polarizer with high extinction ratio were used. The polarizer placed between the photodiode and the iPad is rotated with 10° step. Results presented Fig. 8 show that the response follows Malus’ law with a resulting DoP equal to 0.92. Since the tablet was vertically placed using portrait orientation it results that the light provided by the iPad is polarized according along the major axis.

Fig. 8 Response of a photodiode associated with a rotating polarizing filter placed in front of an iPad.

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5.2.2. Validation of the proposed method

18 images were taken with different orientations of the tablet. All images were then processed according to the algorithm described in Fig. 5. Assuming that the camera was already calibrated using the conventional method with a rotating linear polarizer, the orientation given thanks to the geometric calibration can be compared to the angle of polarization measured by the camera (Fig. 9). As expected the precision regarding the estimation of the angle of polarization computed from the geometric calibration is directly related to the misalignment between the optical axis of the camera and the normal of the iPad provided by the tilt angle. Therefore, only images with low tilt angle must be taken into account to perform the polarization calibration step.

Fig. 9 Comparison between the AoP measured by the camera and the orientation of the Ipad. The top of the figure represents the absolute difference between the AoP estimated by the camera and the orientation of the iPad derived from the geometric calibration. The bottom of the figure represents tilt angle of the iPad. The frame numbers indicated in red were removed for the calibration process.

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Figure 10 presents the polarization calibration results assuming that the orientation of the tablet is provided according to our proposed method through the geometric calibration step. The sinusoid approximations were also performed for every corresponding pixels (belonging to a white square). The error bars represent the standard deviation for the average computation of the sinusoid parameters. The small error in approximating the three sinusoids illustrates how relevant is our method. Contrary to method based on the use of a rotating polarizer, this technique does not rely on precise orientation of the tablet since its pose is directly estimated thanks to the geometric calibration process. Once several images with different orientation of the tablet are acquired, the process is automatic and give both geometric and polarization calibration results. To ensure an efficient calibration for both cases, the tablet must be rotated with various poses: orthofrontal poses for the polarization calibration and with higher tilt angle for geometric calibration. The accuracy of the orientation estimation of the chessboard is directly related to the extrinsic parameters estimation([22]). From our experiment, a precision of around 14 arcmin is obtained.

Fig. 10 Calibration using an iPad results: approximation of the sinusoid parameters during the calibration process. Error bars are at ±1 standard deviation and err represents the approximation error.

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6. Conclusions

In this paper a complete framework to calibrate a three-CCD polarimetric camera is presented. The three polarizers are not always oriented to a precise angle and can have different degree of polarization or average transmittance characteristics leading to wrong estimation of the polarization parameters measured by the camera. Estimating only the orientation of the polarizers is not sufficient and the estimation of the relative values of the degree of polarization and average transmittance between the three polarizers are required. This has been demonstrated using a linearly polarized light source with a precise orientation placed in front of the camera. Even if the real settings of the polarizers can’t be estimated because no prior information regarding the light intensity and the degree of polarization are used we show that estimating only their relative values can significantly improve the polarization images quality. Finally, the proposed method is extended to a very convenient technique using a tablet instead of a precisely oriented linearly polarized light source. The tablet provides both a partially linearly polarized light source and a chessboard to perform in the meantime the polarization calibration and the geometric calibration tasks. The tablet placed in front of the camera is rotated by hand and from the images acquired the polarization calibration matrix and the intrinsic parameters of the camera are directly extracted. This original calibration framework for three-CCD polarimetric cameras will be useful for embedded applications in robotics since no specific optical components are required.

Acknowledgments

This work is part from a project entitled VIPeR (Polarimetric Vision Applied to Robotics Navigation) funded by the French National Research Agency ANR-15-CE22-0009-VIPeR.

References and links

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		AoP deviation		DoP estimation		Intensity
		mean	std	mean	Std	mean	std
original settings	no calibration	8.0°	5.10	0.81	0.169	165.1	22.5
	calibrated angles only θ_i	7.4°	4.63	0.82	0.186	163.7	20.2
	(1) θ_i, T₁ = 0.5, $P_{1} = 1$	0.1°	0.13	0.80	0.005	216.1	2.09
	(2) θ_i, maxT_i = 0.5, $\max P_{i} = 1$	0.1°	0.13	0.82	0.005	212.1	2.06
	(3) θ_i, maxT_i = 0.5, ρ = 1	0.1°	0.13	1.00	0.006	173.7	1.68

Handy method to calibrate division-of-amplitude polarimeters for the first three Stokes parameters

Abstract

1. Introduction

2. Three-CCD polarimetric camera model

3. Calibration using a polarized light source

4. Experimental results

5. Calibration method using a tablet

5.1. Methodology

5.1.1. Automatic corners detection

5.1.2. Geometric calibration

5.1.3. Extraction of relevant data

5.2. Experiments

5.2.1. Tablet properties

5.2.2. Validation of the proposed method

6. Conclusions

Acknowledgments

References and links

Cited By

Figures (10)

Tables (1)

Equations (16)

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