Abstract

A division of focal plane (DoFP) polarimeter includes an array of polarized pixels. The response characteristics of polarized pixels are directly affected by inherent defects of a DoFP polarimeter. Correspondingly, the response characteristics are crucial to correction of the inherent defects. However, research on the response characteristics is rarely reported. Therefore, this paper proposes a pixel response model for a DoFP polarimeter. The response model combines the response characteristics of a traditional photoelectric imager and a micro-polarizer array. The proposed model includes six input parameters. They are the major polarization responsivity, minor polarization responsivity, polarization orientation, exposure time, conversion gain, and gamma correction. An experimental setup is constructed to measure the response of a DoFP polarimeter. The proposed model is evaluated by comparing the calculated results and the measured results. The compared results under different artificial parameters show that the each average root-mean-square error value is less than one gray value, which proves the validity of the proposed model.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1. INTRODUCTION

Polarization of light is another important property in addition to intensity and spectrum. Polarization imaging can obtain physical information resulting from the “asymmetry” of processes such as reflection, refraction, absorption, and scattering. Polarization imaging captures the polarization information of the scene. It is also widely used in many scientific and engineering fields, such as atmospheric remote sensing [14], biomedical diagnosis [5,6], military navigation [79], and marine environments [1013].

The polarimeters are mainly divided into four types. They are division of time (DoT), division of amplitude (DoAM), division of aperture (DoAP), and division of focal plane (DoFP) [14,15]. The DoT polarimeter needs to rotate a polarizer. It is not suitable for observing the motion scene and is not suitable for working on motion platforms. The optical system of a DoAM has a complicated structure and large volume. The polarization images outputted from different channels of a DoAM polarimeter require pixel-level registration, which easily causes spatial registration errors [16]. The DoAP polarimeter has a sophisticated optical path design, high cost, and low reliability. Additionally, for the DoAP polarimeter, different regions of a polarized image require strictly pixel-level registration. A DoFP polarimeter is manufactured by placing a micro-polarizer array directly onto the focal plane array (FPA) of a detector. Each element of a DoFP polarimeter is a polarized pixel. Compared with the other three types of polarimeters, the DoFP polarimeter has prominent merits, such as snapshot imaging, compact structure, low power consumption, high transmission efficiency, and high extinction ratio [17,18].

However, existing DoFP polarimeters suffer from inherent defects. One is instantaneous field of view (IFoV) errors, which are caused by neighboring pixels with different polarization orientations. The other is non-uniformity errors, which are caused by manufacturing and assembly errors of a micro-polarizer array as well as detector noise [14,1921].

Currently, those inherent defects seriously affect the performance of DoFP polarimeters and limit their application. Accordingly, those inherent defects of a DoFP polarimeter have received much attention [15,19,2224]. The DoFP polarimeter includes an array of polarized pixels. The response characteristics of polarized pixels are directly affected by inherent defects of DoFP polarimeters and are crucial to correction of the inherent defects. Therefore, it is necessary to explore the performance and response characteristics of a single polarized pixel. In 2018, Feng et al. proposed a characteristic model, which characterizes the optical and electronic performance of a polarized pixel by three parameters [25]. They are major polarization responsivity, minor polarization responsivity, and polarization orientation [25,26]. However, the response characteristics of polarized pixels have been rarely reported. Therefore, this paper proposes a pixel response model for a DoFP polarimeter. Our model considers major and minor polarization responsivity of a polarized pixel, polarization orientation, and dark offset, and it also takes the exposure time, gain, and gamma conversion parameters of a DoFP polarimeter into consideration.

2. RESPONSE MODEL OF POLARIZED PIXELS

A. Pixel Response Characteristics of Traditional Photoelectric Imager

Figure 1 shows the signal transmission process of the photoelectric imaging system. After the incident light is transmitted through the optical components, it is irradiated on the photosensitive surface of the detector. The photosensitive surface of the detector receives the radiant energy of the optical signal. According to the internal photoelectric effect, the incident radiant energy is converted into an analog electrical signal. Then, this signal goes through the different systems’ function requirements or through various signal processing circuits for amplification, correction, and compensation. The detection signal is processed. Finally, the analog signal processed is converted into a digital signal by the analog-to-digital conversion, and then is outputted and displayed.

 figure: Fig. 1.

Fig. 1. Signal transmission process of the photoelectric imaging system. A/D, analog-to-digital.

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B. Response Model for Polarized Pixels of DoFP Polarimeter

The core of a DoFP polarimeter is the polarization detector. As shown in Fig. 2, the polarization detector is accomplished by integrating a micro-polarizer array onto the FPA of a photodetector. Four adjacent pixels arranged $2\times2$ form a super-pixel. Those four pixels in the super-pixel have different polarization orientations [27,28].

 figure: Fig. 2.

Fig. 2. Structure of the polarization detector.

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In a DoFP polarimeter, the light passing through the micro-polarizer is irradiated to the exposed area of the detector. During the exposure period, the photosensitive array stores the charge. The number of photons received by the detector is controlled by the exposure time. The polarimeter increases the signal-to-noise ratio (SNR) when increasing the exposure time. The SNR is an important performance index of the photoelectric imaging system and is described by photons in the visible light system [29,30]. Choosing the right exposure time can acquire better images. The optical energy received by the photoelectric imaging system is proportional to the exposure time of the optical system.

Based on the traditional photoelectric imaging process, the optical signal received by the photosensor of the detector is converted into an electrical signal. The conversion characteristics of this process are usually expressed in terms of photoelectric sensitivity, which is defined as the brightness gain when the imaging device is used to enhance the visible light image. Changing the system gain can change the sensitivity of the photosensitive device to light. High sensitivity is sensitive to low light and noise signals. The SNR is small, so there is more noise in high sensitivity. Therefore, increasing the gain can get a brighter image, but it increases noise.

The gamma correction circuit is used to correct nonlinear phenomena during photoelectric conversion and transmission. The basic form of gamma correction of traditional pixels is given by

$$S = c \cdot {r^\gamma },$$
where $c$ is a normal number (generally 1), $ \gamma $ is the gamma correction parameter, $r$ is the input gray level, and $S$ is the gray output stage. If the offset is considered (the output when the input is 0),
$$S = c{(r + \varepsilon )^\gamma }.$$
The above equation shows that when $ \gamma = 1 $, the input and output are linear. The curve generated by the value of $ \gamma \gt 1 $ (darkening) and the curve generated by the value of $ \gamma \lt 1 $ (brightening) are distributed on both sides of $ \gamma = 1 $.

Based on theoretical analysis, take $(i,j)$ polarized pixels as an example. As shown in Fig. 3, a linearly polarized light that has an incident light intensity of $I$ and oriented at $ {\varphi _Z} $ is irradiated to the FPA. Due to manufacturing errors, the major transmittance of the micro-polarizer is less than 100%, and the minor transmittance is greater than 0% [25]. According to Malus’s law, the light radiation after passing through a single micro-polarizer is given by

$$\begin{split}{L_{\rm out}} = \mu _{(i,j)}^\parallel I \cdot \cos {({\varphi _z} - {\theta _{(i,j)}})^2} + \mu _{(i,j)}^ \bot I \cdot \sin {({\varphi _z} - {\theta _{(i,j)}})^2},\end{split}$$
where $ \mu _{(i,j)}^\parallel $ and $ \mu _{(i,j)}^ \bot $, respectively, represent the major and minor transmittances of the micro-polarizer, and $ {\theta _{(i,j)}} $ denotes the polarization orientation relative to a reference [25,26]. We call the exposure time $T$ as the reference exposure time. According to the conversion characteristics of incident light, shown in Fig. 4, the analog output of the polarized pixel at the reference exposure time is expressed as
$${d_{(i,j,z)}} = g \cdot {L_{\rm out}} + {b_{(i,j)}},$$
where $g$ represents the conversion coefficient (gain) of the polarized pixel in the system, $ {d_{(i,j,z)}} $ is the output electrical signal, and $ {b_{(i,j)}} $ is the dark offset of the polarization imaging [26]. Based on the results of the above theoretical analysis, some new physical quantities are introduced to characterize the polarized pixel response. Let $ {\beta _t} $ denote the magnification of the exposure time (relative to the reference exposure time), which can change the period that the photosensitive array receives the photons. Let $ {\beta _g} $ denote the magnification of the gain (relative to the reference gain), the physical quantity that acts on the photographic performance of the photosensitive array. The output of the corresponding polarized pixel is expressed as
$$d_{(i,j,z)}^ * = {\beta _g} \cdot {\beta _t} \cdot g \cdot {L_{\rm out}} + {b_{(i,j)}}.$$

 figure: Fig. 3.

Fig. 3. Polarized transmission direction of linearly polarized light and polarized pixels.

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 figure: Fig. 4.

Fig. 4. Conversion diagram of optical radiation signal to electrical signal.

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Substituting Eq. (3) into Eq. (5), we can obtain

$$\begin{split}d_{(i,j,z)}^ * &= {\beta _g} \cdot {\beta _t}\Big\{ g\Big[\mu _{(i,j)}^\parallel I \cdot \cos {({\varphi _z} - {\theta _{(i,j)}})^2} \\&\quad+ \mu _{(i,j)}^ \bot I \cdot \sin {({\varphi _z} - {\theta _{(i,j)}})^2}\Big]\Big\} + {b_{(i,j)}}.\end{split}$$
Let
$$\eta _{(i,j)}^\parallel = g \cdot \mu _{(i,j)}^\parallel ,$$
$$\eta _{(i,j)}^ \bot = g \cdot \mu _{(i,j)}^ \bot .$$
The expression can be rewritten as
$$\begin{split}d_{(i,j,z)}^ * &= {\beta _g} \cdot {\beta _t}\Big[\eta _{(i,j)}^\parallel I \cdot \cos {({\varphi _z} - {\theta _{(i,j)}})^2}\\&\quad + \eta _{(i,j)}^ \bot I \cdot \sin {({\varphi _z} - {\theta _{(i,j)}})^2}\Big] + {b_{(i,j)}},\end{split}$$
where $ \eta _{(i,j)}^\parallel $ and $ \eta _{(i,j)}^ \bot $, respectively, represent the major and minor polarization responsivity of the reference gain. Equation (9) gives the response from the incident light to the analog signal output. Compared with the traditional gamma correction, we bring Eq. (9) into Eq. (2) to get the final polarization pixel response model:
$$\begin{split}{S_{(i,j,z)}} &= \Big\{ {\beta _g} \cdot {\beta _t}\Big[\eta _{(i,j)}^\parallel I \cdot \cos {({\varphi _z} - {\theta _{(i,j)}})^2}\\&\quad + \eta _{(i,j)}^ \bot I \cdot \sin {({\varphi _z} - {\theta _{(i,j)}})^2}\Big] + {b_{(i,j)}}\Big\} ^\gamma .\end{split}$$
According to the following mathematical relations,
$$\cos {({\varphi _z} - {\theta _{(i,j)}})^2}=\frac{{1+\cos [2({\varphi _z} - {\theta _{(i,j)}})]}}{2},$$
$$\sin {({\varphi _z} - {\theta _{(i,j)}})^2}=\frac{{1 - \cos [2({\varphi _z} - {\theta _{(i,j)}})]}}{2}.$$
Equation (10) can be rewritten as
$$\begin{split}{S_{(i,j,z)}} &= \bigg\{ \frac{1}{2}{\beta _g} \cdot {\beta _t} \cdot I\left[(\eta _{(i,j)}^\parallel + \eta _{(i,j)}^ \bot )+ (\eta _{(i,j)}^\parallel - \eta _{(i,j)}^ \bot )\right.\kern-2pc\\[-3pt]&\quad \left.\times\cos (2{\varphi _z} - 2{\theta _{(i,j)}})\vphantom{\left[(\eta _{(i,j)}^\parallel + \eta _{(i,j)}^ \bot )+ (\eta _{(i,j)}^\parallel - \eta _{(i,j)}^ \bot )\right.}\right] + {b_{(i,j)}}\bigg\} ^\gamma ,\kern-2pc\end{split}$$
$$\eta _{(i,j)}^ + = \frac{1}{2}(\eta _{(i,j)}^\parallel + \eta _{(i,j)}^ \bot ),$$
$$\eta _{(i,j)}^ - = \frac{1}{2}(\eta _{(i,j)}^\parallel - \eta _{(i,j)}^ \bot ),$$
then
$$\begin{split}{S_{(i,j,z)}} &= \Big\{ \Big[{\beta _g} \cdot {\beta _t} \cdot I(\eta _{(i,j)}^ + + \eta _{(i,j)}^ - \cos 2{\varphi _z}\cos 2{\theta _{(i,j)}}\\[-2pt]&\quad + \eta _{(i,j)}^ - \sin 2{\varphi _z}\sin 2{\theta _{(i,j)}})\Big] + {b_{(i,j)}}\Big\} ^\gamma ,\end{split}$$
where $ \gamma $ is the nonlinear processing of the pixel. For given a certain polarization pixel $ (i,j) $, the parameters of $ \eta _{(i,j)}^ + $, $ \eta _{(i,j)}^ - $, $ {\theta _{(i,j)}} $, and $ {b_{(i,j)}} $ are invariable and unknown. However, the parameters of exposure time, gain, and gamma correction are artificially set, so the parameters $ {\beta _g} $, $ {\beta _t} $, and $ \gamma $ are known in the model. In Eq. (16), we select the calibration of the polarization response model with reference exposure time, reference gain, and $ \gamma = 1 $. By changing the orientation $ {\varphi _z} $ of the linearly polarized light, the polarization camera will output different response values $ {S_{(i,j,z)}} $. The parameter of $ {b_{(i,j)}} $ is the dark offset. It can be measured by covering the lens of a camera. Equation (16) can be written as
$$\begin{split}&\left[ {\begin{array}{*{20}{c}}{\begin{array}{*{20}{c}}{\begin{array}{*{20}{c}}{S_{(i,j,z)}^1}\\{S_{(i,j,z)}^2}\end{array}}\\ \vdots \end{array}}\\{S_{(i,j,z)}^n}\end{array}} \right] - \left[ {\begin{array}{*{20}{c}}{\begin{array}{*{20}{c}}{\begin{array}{*{20}{c}}{b_{(i,j)}^1}\\{b_{(i,j)}^2}\end{array}}\\ \vdots \end{array}}\\{b_{(i,j)}^n}\end{array}} \right]\\&\quad = I\left[ {\begin{array}{*{20}{c}}{\begin{array}{*{20}{c}}1\\1\\ \vdots \\1\end{array}}&{\begin{array}{*{20}{c}}{\cos 2{\varphi _1}}\\{\cos 2{\varphi _2}}\\ \vdots \\{\cos 2{\varphi _n}}\end{array}}&{\begin{array}{*{20}{c}}{\sin 2{\varphi _1}}\\{\sin 2{\varphi _2}}\\ \vdots \\{\sin 2{\varphi _n}}\end{array}}\end{array}} \right]\left[ {\begin{array}{*{20}{c}}{\eta _{(i,j)}^ + }\\{\eta _{(i,j)}^ - \cos 2{\theta _{(i,j)}}}\\{\eta _{(i,j)}^ - \sin 2{\theta _{(i,j)}}}\end{array}} \right].\end{split}$$
Let
$$\left[ {\begin{array}{*{20}{c}}{\begin{array}{*{20}{c}}{\begin{array}{*{20}{c}}{y_{(i,j)}^1}\\{y_{(i,j)}^2}\end{array}}\\ \vdots \end{array}}\\{y_{(i,j)}^n}\end{array}} \right] = \left[ {\begin{array}{*{20}{c}}{\begin{array}{*{20}{c}}{\begin{array}{*{20}{c}}{S_{(i,j,z)}^1}\\{S_{(i,j,z)}^2}\end{array}}\\ \vdots \end{array}}\\{S_{(i,j,z)}^n}\end{array}} \right] - \left[ {\begin{array}{*{20}{c}}{\begin{array}{*{20}{c}}{\begin{array}{*{20}{c}}{b_{(i,j)}^1}\\{b_{(i,j)}^2}\end{array}}\\ \vdots \end{array}}\\{b_{(i.j)}^n}\end{array}} \right],$$
$$\left[ {\begin{array}{*{20}{c}}{{q_1}}\\{{q_2}}\\{{q_3}}\end{array}} \right] = \left[ {\begin{array}{*{20}{c}}{\eta _{(i,j)}^ + }\\{\eta _{(i,j)}^ - \cos 2{\theta _{(i,j)}}}\\{\eta _{(i,j)}^ - \sin 2{\theta _{(i,j)}}}\end{array}} \right].$$
By means of the least-squares method,
$$\left[ {\begin{array}{*{20}{c}}{{q_1}}\\{{q_2}}\\{{q_3}}\end{array}} \right] = \frac{1}{I}{\left[ {\begin{array}{*{20}{c}}{\begin{array}{*{20}{c}}1\\1\\ \vdots \\1\end{array}}&{\begin{array}{*{20}{c}}{\cos 2{\varphi _1}}\\{\cos 2{\varphi _2}}\\ \vdots \\{\cos 2{\varphi _n}}\end{array}}&{\begin{array}{*{20}{c}}{\sin 2{\varphi _1}}\\{\sin 2{\varphi _2}}\\ \vdots \\{\sin 2{\varphi _n}}\end{array}}\end{array}} \right]^\dagger }\left[ {\begin{array}{*{20}{c}}{\begin{array}{*{20}{c}}{\begin{array}{*{20}{c}}{y_{(i,j)}^1}\\{y_{(i,j)}^2}\end{array}}\\ \vdots \end{array}}\\{y_{(i,j)}^n}\end{array}} \right].$$
Then,
$${\theta _{(i,j)}} = \frac{1}{2}\,\arctan \left(\frac{{{q_3}}}{{{q_2}}}\right),$$
$$\eta _{(i,j)}^\parallel = \frac{{{q_1}}}{2} + \frac{{{q_2}}}{{2\cos 2{\theta _{(i,j)}}}},$$
$$\eta _{(i,j)}^ \bot = \frac{{{q_1}}}{2} - \frac{{{q_2}}}{{2\cos 2{\theta _{(i,j)}}}},$$
where the symbol $\dagger $ represents a pseudo-inverse operator. Equation (20) can solve the values of $ {q_1},{q_2} $, and $ {q_3} $. Then, by using Eqs. (21)–(23), the polarization orientation $ {\theta _{(i,j)}} $, the major polarization responsivity, and the minor polarization responsivity of the polarization pixel $ (i,j) $ are obtained. Therefore, given a linearly polarized light, the output response of the polarized pixel can be obtained by the calibrated parameters and the artificially set parameters.

This paper proposes a polarized pixel response model that takes into account the three parameters of exposure time, gain, and gamma correction, as shown in Eq. (10).

3. EXPERIMENTAL RESULTS AND ANALYSIS

A. Experimental Setup

Figure 5 shows the optical diagram of our experimental setup. The LSH-T150 tungsten-halogen light source and the LSP-T150 power supply for the tungsten-halogen lamp are used together to provide a highly stable current source. The fluctuation of luminous ${\rm flux} \lt = 1\% $. This device produces a linear white light source (HSIA-LS-TS-30), and then through the integrating sphere (US-120-SF) outputs a uniform light source. The clear aperture diameter of the integrating sphere is 101.6 mm, the internal reflectivity is more than 98%, and the uniformity is greater than 98%. It is followed by a motorized precision rotary stage (changing the polarization orientation of the linearly polarized light) to obtain uniform linearly polarized light. The device is controlled by a DC servo motor controller, which can obtain a precise and controllable polarization light source. The specifications of each device are shown in Table 1, and the experimental setup is shown in Fig. 6.

 figure: Fig. 5.

Fig. 5. Optical diagram of our experimental setup.

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Table 1. Experimental Setup Model

 figure: Fig. 6.

Fig. 6. Constructed experimental setup.

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The pixel resolution of the DoFP polarized camera is $2448 \times 2048$. To reduce the influence of time noise, we use multiple sets of measurements to calculate the average value for each group. The proposed model is validated by comparing the errors between the measured values and the calculated values.

We choose root-mean-square errors (RMSEs) to characterize errors of experimental data. RMSE reflects the deviation between the measured values and the calculated values. The RMSE is calculated by

$${\rm RMSE} = \sqrt {\frac{1}{n}\sum\limits_{i = 1}^n {[{y_i} - \phi ({x_i})} {]^2}} ,$$
Where $ {y_i} $ represents the measured value, $ \phi ({x_i}) $ represents the calculated value, and $n$ refers to the number of measurements.

B. Experimental Results

To validate the proposed model, we conducted two experimental tasks on the experimental platform shown in Fig. 6. In the experiment, we collected the polarization information, which is displayed as an 8 bit gray image, and the range of its gray value is between 0 and 255. The error assessment in the paper is based on the actual measurement value.

1. First Task

The first task is to validate the response model under the reference condition.

The experimental data used to calculate model parameters and used to test the model came from different sampled points. The sampled data of 14 groups $ {\varphi _z}(0^\circ ,20^\circ ,40^\circ ,...,260^\circ ) $ are selected as training groups for calculated unknown parameters. Then, the sampled data of 14 groups $ {\varphi _z}(10^\circ ,30^\circ ,50^\circ ,...,270^\circ ) $ are selected as the test groups to prove the correctness of the model. For the artificial parameters, the exposure time is set to 3500 µs, the gain is set to 1, and the gamma correction value is set $\gamma = 1$. The parameters in the model (reference orientation, major polarization, and minor polarization) are calculated under reference conditions. The procedure can be described as follows,

Step 1: Produce a linearly polarized light whose polarization orientation $ {\varphi _z}(0^\circ ,10^\circ ,20^\circ ,...,270^\circ ) $ is changed by a motorized precision rotator.

Step 2: Under each linearly polarized light, we collect multiple groups of polarization images and calculate the average image of each group.

Step 3: The dark offset $ {b_{(i,j)}} $ is obtained by covering the camera lens under the reference condition.

Step 4: By Eqs. (17)–(23), we can obtain the polarization orientations relative to a reference orientation $ {\theta _{(i,j)}} $, the major polarization responsivity, and the minor polarization responsivity of the polarization pixel $ (i,j) $.

Figure 7 shows the fitting results of a super-pixel (0°, 45°, 90°, 135°) response model under the reference conditions. The corresponding RMSE values are $ {{\rm RMSE}_{0^\circ }} = 0.2596 $, $ {{\rm RMSE}_{45^\circ }} = 0.3766 $, $ {{\rm RMSE}_{90^\circ }} = 0.3791 $, and $ {{\rm RMSE}_{135^\circ }} = 0.2389 $.

 figure: Fig. 7.

Fig. 7. Response of a super-pixel under reference conditions.

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 figure: Fig. 8.

Fig. 8. Comparison between calculated values and measured values under different exposure times and different linearly polarized light.

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2. Second Task

The second task is to validate the proposed model in seven cases.

In this section, the test data $ {\varphi _z}(0^\circ ,10^\circ ,20^\circ ,...,270^\circ ) $ in the seven cases is used as the training groups and the sampled data under the reference conditions as the test groups.

Case 1: Under different exposure times and different linearly polarized light. We adjust the exposure time $T$ (related to $ {\beta _t} $) of the camera at a step of 500 µs relative to the reference exposure time. The exposure time is, respectively, adjusted to 3500 µs, 4000 µs, 4500 µs, …, 7500 µs, and the actual values are 3499 µs, 3998 µs, 4503 µs, …, 7502 µs. At each exposure time $T$, by changing the incident linearly polarized light, the output data are measured, and the model response curve is fitted. Figure 8 shows the comparison between calculated values and measured values under different exposure times and different linearly polarized light. In the legend, $T$ represents the exposure time, and its unit is microseconds. Table 2 gives the RMSE values between calculated results and measured results at different exposure times.

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Table 2. RMSE Values between Calculated Results and Measured Results at Different Exposure Times

Case 2: Under different exposure times and given linearly polarized light. Given the pixel point (1024, 1024) and the linearly polarized light ($ {\varphi _z} = 45 $), under each changed exposure time, the calculated value by our model and the measured value are compared. The compared results in Fig. 9 show that the calculated values are close to the measured values under different exposure times. Their difference is evaluated by RMSE. The RMSE value is 1.0755. The response range of the polarized pixel is [0,255]. This RMSE value of 1.0755 is less than the response maximum of the polarized pixel. The response of a polarized pixel is proximately proportional to its exposure time.

 figure: Fig. 9.

Fig. 9. Comparison between calculated values and measured values under different exposure times and given linearly polarized light.

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Case 3: Under different gains and different linearly polarized light. We adjust the gain $g$ (related to $ {\beta _g} $) of the camera to 1 dB, 2 dB, …, 8 dB, respectively. As shown in Fig. 10, by changing the incident linearly polarized light at each gain, the output data are measured, and the model response curve is fitted. In the legend, $g$ represents the gain, and its unit is decibels. Table 3 gives the RMSE values between calculated results and measured results at different gains.

 figure: Fig. 10.

Fig. 10. Comparison between calculated values and measured values under different gains and different linearly polarized light.

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Table 3. RMSE Values between Calculated Results and Measured Results at Different Gains

 figure: Fig. 11.

Fig. 11. Comparison between calculated values and measured values under different gains and given linearly polarized light.

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Case 4: Under different gains and given linearly polarized light. According to the inherent characteristics of the camera, in image sensor applications, the gain (dB) is defined as shown in the following:

$$20\log (y) = (x)\,\,{\rm dB},$$
where $x$ is the gain 1 dB, 2 dB, …, 8 dB, and $y$ is the converted ratio data.

Therefore, when verifying the relationship between gain and output value, we convert 1–8 dB into conventional ratio data. Given linearly polarized light and pixel point (1024, 1024) are the same as that in case 1. Under each changed gain, the measured value and the calculated value by our model are compared. The compared results are shown in Fig. 11. The compared results show that the calculated values by our model are close to the measured values under different gains, and the RMSE is 0.6324. This RMSE value of 0.6324 is less than the response maximum of the polarized pixel. The distribution of the theoretical analysis result is consistent with the measured value.

Case 5: Under different gamma values and different linearly polarized light. We take gamma values as 0.6, 0.8, …, 1.4. By changing the incident linearly polarized light at each gamma value, the output data are measured, and the model response curve is fitted. Due to the inherent characteristics of the camera, the reciprocal of the actual displayed gamma value needs to be calculated, and the gray value of the polarized pixel needs to be normalized. Figure 12 shows the comparison between calculated values and the measured values under different gamma values and different linearly polarized light. Table 4 gives the RMSE values of the corresponding data at different gamma values.

 figure: Fig. 12.

Fig. 12. Comparison between calculated values and measured values under different gamma values and different linearly polarized light.

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Table 4. RMSE Values between Calculated Results and Measured Results at Different Gamma Values

Case 6: Under different gamma values and given linearly polarized light. Given linearly polarized light and pixel point (1024, 1024) are the same as that in case 1. Under each changed gamma value, the measurement value and the calculated value by our model are compared. The compared results are shown in Fig. 13. The compared results show that the calculated values by our model are close to the measured values under different gamma values. Their RMSE value is 0.2988, and this RMSE value is much less than the response maximum of the polarized pixel.

Case 7: Average RMSE values of polarized pixels with same polarization orientation. To further verify the validity of the polarization model, we performed a comprehensive analysis of the data measured. According to structures of the micro-polarizer array, the polarization orientations under each exposure time, gain, and gamma value can be classified into four polarization orientations (0°, 45°, 90°, 135°). Then, for each polarization orientation, the RMSE between measured results and the model calculated results are averaged on the entire micro-polarizer array. The average RMSE values for each polarization orientation are shown in Tables 57. All RMSE values are much less than the response maximum of the polarized pixel. The results show that the calculated values by our model are close to the measured values under different parameters.

 figure: Fig. 13.

Fig. 13. Comparison between calculated values and measured values under different gamma values and given linearly polarized light.

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Tables Icon

Table 5. Average RMSE of Polarized Pixels for Each Polarization Orientation at Different Exposure Times

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Table 6. Average RMSE of Polarized Pixels for Each Polarization Orientation at Different Gains

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Table 7. Average RMSE of Polarized Pixels for Each Polarization Orientation at Different Gamma Values

The experimental results prove the following:

  • (1) Under the reference condition, for a super-pixel, RMSE values between the model calculated value and the actual measured value are less than 1 gray value, and they are much less than the maximum gray value of one pixel. The proposed model is validated under the reference condition. Under different exposure times, gains, gamma values, and average RMSE values between the model calculated value and the actual measured value are less than 1 gray value, which proves the validity of the proposed model.
  • (2) We conduct experiments under different artificial parameters. Those artificial parameters include exposure time, gain, and gamma. The results in Tables 57 show that the average RMSE values of polarized pixels under different artificial parameters are between 0.1813 and 0.5542. The calculated response value of the polarized pixel is close to its actual measured value, which proves the validity of the proposed model under different artificial parameters.

4. CONCLUSIONS

This paper proposes a pixel response model for a DoFP polarimeter. The proposed model includes six input parameters. They are the major polarization responsivity, minor polarization responsivity, polarization orientation, exposure time, conversion gain, and gamma correction. The model’s validity is experimentally validated under different conditions. By this model, the response for a given linearly polarized light is predictable. However, as an extension of the proposed model, the pixel response for elliptical polarized light and partially polarized light deserves further discussion. Additionally, how to apply the proposed model to correct the inherent defects of a DoFP polarimeter also deserves further research.

Funding

Shanxi Provincial Key Research and Development Project (2018ZDXM-GY-091); National Natural Science Foundation of China (61805199); Natural Science Foundation of Shaanxi Province (2018JQ6065); Key Technologies Research and Development Program (2018YFB1309403).

Acknowledgment

The authors would like to thank Associate Professor Liang Nie from Xi’an Technological University for providing the integrating sphere equipment. The authors would also like to thank every anonymous reviewer for his/her detailed comments.

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2. S. Fang, X. Xia, H. Xing, and C. Chen, “Image dehazing using polarization effects of objects and airlight,” Opt. Express 22, 19523–19537 (2014). [CrossRef]  

3. B. Huang, T. Liu, J. Han, and H. Hu, “Polarimetric target detection under uneven illumination,” Opt. Express 23, 23603–23612 (2015). [CrossRef]  

4. R. Liu, D. J. Wang, P. Jia, and X. Che, “Gradient sky scene based nonuniformity correction and local weighted filter based denoising,” Optik 174, 748–756 (2018). [CrossRef]  

5. A. Pierangelo, A. Benali, M.-R. Antonelli, T. Novikova, P. Validire, B. Gayet, and A. De Martino, “Ex-vivo characterization of human colon cancer by Mueller polarimetric imaging,” Opt. Express 19, 1582–1593 (2011). [CrossRef]  

6. J. Qi and D. S. Elson, “Mueller polarimetric imaging for surgical and diagnostic applications: a review,” J. Biophoton. 10, 950–982 (2017). [CrossRef]  

7. J. Duan, Q. Fu, C. Mo, Y. Zhu, and D. Liu, “Review of polarization imaging for international military application,” Proc. SPIE 8908, 890813 (2013) [CrossRef]  

8. F. Goudail and M. Boffety, “Performance comparison of fully adaptive and static passive polarimetric imagers in the presence of intensity and polarization contrast,” J. Opt. Soc. Am. A 33, 1880–1886 (2016). [CrossRef]  

9. L. Guo, E. Cao, G. Gu, X. Hu, W. Qian, M. Wan, and R. Zhao, “Target recognition method based on polarization parameters,” Proc. SPIE 10155, 101552N (2016). [CrossRef]  

10. M. Garcia, C. Edmiston, R. Marinov, A. Vail, and V. Gruev, “Bio-inspired color-polarization imager for real-time in situ imaging,” Optica 4, 1263–1271 (2017). [CrossRef]  

11. F. Goudail and M. Boffety, “Fundamental limits of target detection performance in passive polarization imaging,” J. Opt. Soc. Am. A 34, 506–512 (2017). [CrossRef]  

12. D. B. Chenault, J. P. Vaden, D. A. Mitchell, and E. D. Demicco, “Infrared polarimetric sensing of oil on water,” Mar. Technol. Soc. J. 52, 13–22 (2018). [CrossRef]  

13. S. B. Powell, R. Garnett, J. Marshall, C. Rizk, and V. Gruev, “Bioinspired polarization vision enables underwater geolocalization,” Sci. Adv. 4, eaao6841 (2018). [CrossRef]  

14. Z. Chen, X. Wang, and R. Liang, “Calibration method of microgrid polarimeters with image interpolation,” Appl. Opt. 54, 995–1001 (2015). [CrossRef]  

15. J. Zhang, H. Luo, B. Hui, and Z. Chang, “Image interpolation for division of focal plane polarimeters with intensity correlation,” Opt. Express 24, 20799–20807 (2016). [CrossRef]  

16. E. Compain and B. Drevillon, “Broadband division-of-amplitude polarimeter based on uncoated prisms,” Appl. Opt. 37, 5938–5944 (1998). [CrossRef]  

17. W.-L. Hsu, G. Myhre, K. Balakrishnan, N. Brock, M. Ibn-Elhaj, and S. Pau, “Full-Stokes imaging polarimeter using an array of elliptical polarizer,” Opt. Express 22, 3063–3074 (2014). [CrossRef]  

18. G. Myhre, W.-L. Hsu, A. Peinado, C. LaCasse, N. Brock, R. A. Chipman, and S. Pau, “Liquid crystal polymer full-Stokes division of focal plane polarimeter,” Opt. Express 20, 27393–27409 (2012). [CrossRef]  

19. J. Zhang, H. Luo, B. Hui, and Z. Chang, “Non-uniformity correction for division of focal plane polarimeters with a calibration method,” Appl. Opt. 55, 7236–7240 (2016). [CrossRef]  

20. A. Ahmed, X. Zhao, V. Gruev, J. Zhang, and A. Bermak, “Residual interpolation for division of focal plane polarization image sensors,” Opt. Express 25, 10651–10662 (2017). [CrossRef]  

21. M. Garcia, T. Davis, S. Blair, N. Cui, and V. Gruev, “Bioinspired polarization imager with high dynamic range,” Optica 5, 1240–1246 (2018). [CrossRef]  

22. H. Fei, F.-M. Li, W.-C. Chen, R. Zhang, and C.-S. Chen, “Calibration method for division of focal plane polarimeters,” Appl. Opt. 57, 4992–4996 (2018). [CrossRef]  

23. S. B. Powell and V. Gruev, “Calibration methods for division-of-focal-plane polarimeters,” Opt. Express 21, 21039–21055 (2013). [CrossRef]  

24. D. Vorobiev, Z. Ninkov, N. Brock, and R. West, “On-sky performance evaluation and calibration of a polarization-sensitive focal plane array,” Proc. SPIE 9912, 99125X (2016). [CrossRef]  

25. B. Feng, Z. Shi, H. Liu, L. Liu, Y. Zhao, and J. Zhang, “Polarized-pixel performance model for DoFP polarimeter,” J. Opt. 20, 065703 (2018). [CrossRef]  

26. B. Feng, Z. Shi, H. Liu, Y. Zhao, and J. Liu, “Calibration method for equivalent extinction ratio of polarized pixel in integrated micropolarizer array camera,” Proc. SPIE 10605, 106051B (2017). [CrossRef]  

27. J. Zhang, H. Luo, R. Liang, W. Zhou, B. Hui, and Z. Chang, “PCA-based denoising method for division of focal plane polarimeters,” Opt. Express 25, 2391–2400 (2017). [CrossRef]  

28. S. Roussel, M. Boffety, and F. Goudail, “Polarimetric precision of a micropolarizer grid-based camera in the presence of additive and Poisson shot noise,” Opt. Express 26, 29968–29982 (2018). [CrossRef]  

29. Z. Chen, X. Wang, S. Pacheco, and R. Liang, “Impact of CCD camera SNR on polarimetric accuracy,” Appl. Opt. 53, 7649–7656 (2014). [CrossRef]  

30. H. Sun, D. Wang, C. Chen, K. Long, and X. Sun, “Effect of sensor SNR and extinction ratio on polarimetric imaging error for nanowire-based systems,” Appl. Opt. 57, 7344–7351 (2018). [CrossRef]  

References

  • View by:

  1. E. Namer, S. Shwartz, and Y. Y. Schechner, “Skyless polarimetric calibration and visibility enhancement,” Opt. Express 17, 472–493 (2009).
    [Crossref]
  2. S. Fang, X. Xia, H. Xing, and C. Chen, “Image dehazing using polarization effects of objects and airlight,” Opt. Express 22, 19523–19537 (2014).
    [Crossref]
  3. B. Huang, T. Liu, J. Han, and H. Hu, “Polarimetric target detection under uneven illumination,” Opt. Express 23, 23603–23612 (2015).
    [Crossref]
  4. R. Liu, D. J. Wang, P. Jia, and X. Che, “Gradient sky scene based nonuniformity correction and local weighted filter based denoising,” Optik 174, 748–756 (2018).
    [Crossref]
  5. A. Pierangelo, A. Benali, M.-R. Antonelli, T. Novikova, P. Validire, B. Gayet, and A. De Martino, “Ex-vivo characterization of human colon cancer by Mueller polarimetric imaging,” Opt. Express 19, 1582–1593 (2011).
    [Crossref]
  6. J. Qi and D. S. Elson, “Mueller polarimetric imaging for surgical and diagnostic applications: a review,” J. Biophoton. 10, 950–982 (2017).
    [Crossref]
  7. J. Duan, Q. Fu, C. Mo, Y. Zhu, and D. Liu, “Review of polarization imaging for international military application,” Proc. SPIE 8908, 890813 (2013)
    [Crossref]
  8. F. Goudail and M. Boffety, “Performance comparison of fully adaptive and static passive polarimetric imagers in the presence of intensity and polarization contrast,” J. Opt. Soc. Am. A 33, 1880–1886 (2016).
    [Crossref]
  9. L. Guo, E. Cao, G. Gu, X. Hu, W. Qian, M. Wan, and R. Zhao, “Target recognition method based on polarization parameters,” Proc. SPIE 10155, 101552N (2016).
    [Crossref]
  10. M. Garcia, C. Edmiston, R. Marinov, A. Vail, and V. Gruev, “Bio-inspired color-polarization imager for real-time in situ imaging,” Optica 4, 1263–1271 (2017).
    [Crossref]
  11. F. Goudail and M. Boffety, “Fundamental limits of target detection performance in passive polarization imaging,” J. Opt. Soc. Am. A 34, 506–512 (2017).
    [Crossref]
  12. D. B. Chenault, J. P. Vaden, D. A. Mitchell, and E. D. Demicco, “Infrared polarimetric sensing of oil on water,” Mar. Technol. Soc. J. 52, 13–22 (2018).
    [Crossref]
  13. S. B. Powell, R. Garnett, J. Marshall, C. Rizk, and V. Gruev, “Bioinspired polarization vision enables underwater geolocalization,” Sci. Adv. 4, eaao6841 (2018).
    [Crossref]
  14. Z. Chen, X. Wang, and R. Liang, “Calibration method of microgrid polarimeters with image interpolation,” Appl. Opt. 54, 995–1001 (2015).
    [Crossref]
  15. J. Zhang, H. Luo, B. Hui, and Z. Chang, “Image interpolation for division of focal plane polarimeters with intensity correlation,” Opt. Express 24, 20799–20807 (2016).
    [Crossref]
  16. E. Compain and B. Drevillon, “Broadband division-of-amplitude polarimeter based on uncoated prisms,” Appl. Opt. 37, 5938–5944 (1998).
    [Crossref]
  17. W.-L. Hsu, G. Myhre, K. Balakrishnan, N. Brock, M. Ibn-Elhaj, and S. Pau, “Full-Stokes imaging polarimeter using an array of elliptical polarizer,” Opt. Express 22, 3063–3074 (2014).
    [Crossref]
  18. G. Myhre, W.-L. Hsu, A. Peinado, C. LaCasse, N. Brock, R. A. Chipman, and S. Pau, “Liquid crystal polymer full-Stokes division of focal plane polarimeter,” Opt. Express 20, 27393–27409 (2012).
    [Crossref]
  19. J. Zhang, H. Luo, B. Hui, and Z. Chang, “Non-uniformity correction for division of focal plane polarimeters with a calibration method,” Appl. Opt. 55, 7236–7240 (2016).
    [Crossref]
  20. A. Ahmed, X. Zhao, V. Gruev, J. Zhang, and A. Bermak, “Residual interpolation for division of focal plane polarization image sensors,” Opt. Express 25, 10651–10662 (2017).
    [Crossref]
  21. M. Garcia, T. Davis, S. Blair, N. Cui, and V. Gruev, “Bioinspired polarization imager with high dynamic range,” Optica 5, 1240–1246 (2018).
    [Crossref]
  22. H. Fei, F.-M. Li, W.-C. Chen, R. Zhang, and C.-S. Chen, “Calibration method for division of focal plane polarimeters,” Appl. Opt. 57, 4992–4996 (2018).
    [Crossref]
  23. S. B. Powell and V. Gruev, “Calibration methods for division-of-focal-plane polarimeters,” Opt. Express 21, 21039–21055 (2013).
    [Crossref]
  24. D. Vorobiev, Z. Ninkov, N. Brock, and R. West, “On-sky performance evaluation and calibration of a polarization-sensitive focal plane array,” Proc. SPIE 9912, 99125X (2016).
    [Crossref]
  25. B. Feng, Z. Shi, H. Liu, L. Liu, Y. Zhao, and J. Zhang, “Polarized-pixel performance model for DoFP polarimeter,” J. Opt. 20, 065703 (2018).
    [Crossref]
  26. B. Feng, Z. Shi, H. Liu, Y. Zhao, and J. Liu, “Calibration method for equivalent extinction ratio of polarized pixel in integrated micropolarizer array camera,” Proc. SPIE 10605, 106051B (2017).
    [Crossref]
  27. J. Zhang, H. Luo, R. Liang, W. Zhou, B. Hui, and Z. Chang, “PCA-based denoising method for division of focal plane polarimeters,” Opt. Express 25, 2391–2400 (2017).
    [Crossref]
  28. S. Roussel, M. Boffety, and F. Goudail, “Polarimetric precision of a micropolarizer grid-based camera in the presence of additive and Poisson shot noise,” Opt. Express 26, 29968–29982 (2018).
    [Crossref]
  29. Z. Chen, X. Wang, S. Pacheco, and R. Liang, “Impact of CCD camera SNR on polarimetric accuracy,” Appl. Opt. 53, 7649–7656 (2014).
    [Crossref]
  30. H. Sun, D. Wang, C. Chen, K. Long, and X. Sun, “Effect of sensor SNR and extinction ratio on polarimetric imaging error for nanowire-based systems,” Appl. Opt. 57, 7344–7351 (2018).
    [Crossref]

2018 (8)

R. Liu, D. J. Wang, P. Jia, and X. Che, “Gradient sky scene based nonuniformity correction and local weighted filter based denoising,” Optik 174, 748–756 (2018).
[Crossref]

D. B. Chenault, J. P. Vaden, D. A. Mitchell, and E. D. Demicco, “Infrared polarimetric sensing of oil on water,” Mar. Technol. Soc. J. 52, 13–22 (2018).
[Crossref]

S. B. Powell, R. Garnett, J. Marshall, C. Rizk, and V. Gruev, “Bioinspired polarization vision enables underwater geolocalization,” Sci. Adv. 4, eaao6841 (2018).
[Crossref]

M. Garcia, T. Davis, S. Blair, N. Cui, and V. Gruev, “Bioinspired polarization imager with high dynamic range,” Optica 5, 1240–1246 (2018).
[Crossref]

H. Fei, F.-M. Li, W.-C. Chen, R. Zhang, and C.-S. Chen, “Calibration method for division of focal plane polarimeters,” Appl. Opt. 57, 4992–4996 (2018).
[Crossref]

B. Feng, Z. Shi, H. Liu, L. Liu, Y. Zhao, and J. Zhang, “Polarized-pixel performance model for DoFP polarimeter,” J. Opt. 20, 065703 (2018).
[Crossref]

S. Roussel, M. Boffety, and F. Goudail, “Polarimetric precision of a micropolarizer grid-based camera in the presence of additive and Poisson shot noise,” Opt. Express 26, 29968–29982 (2018).
[Crossref]

H. Sun, D. Wang, C. Chen, K. Long, and X. Sun, “Effect of sensor SNR and extinction ratio on polarimetric imaging error for nanowire-based systems,” Appl. Opt. 57, 7344–7351 (2018).
[Crossref]

2017 (6)

2016 (5)

2015 (2)

2014 (3)

2013 (2)

S. B. Powell and V. Gruev, “Calibration methods for division-of-focal-plane polarimeters,” Opt. Express 21, 21039–21055 (2013).
[Crossref]

J. Duan, Q. Fu, C. Mo, Y. Zhu, and D. Liu, “Review of polarization imaging for international military application,” Proc. SPIE 8908, 890813 (2013)
[Crossref]

2012 (1)

2011 (1)

2009 (1)

1998 (1)

Ahmed, A.

Antonelli, M.-R.

Balakrishnan, K.

Benali, A.

Bermak, A.

Blair, S.

Boffety, M.

Brock, N.

Cao, E.

L. Guo, E. Cao, G. Gu, X. Hu, W. Qian, M. Wan, and R. Zhao, “Target recognition method based on polarization parameters,” Proc. SPIE 10155, 101552N (2016).
[Crossref]

Chang, Z.

Che, X.

R. Liu, D. J. Wang, P. Jia, and X. Che, “Gradient sky scene based nonuniformity correction and local weighted filter based denoising,” Optik 174, 748–756 (2018).
[Crossref]

Chen, C.

Chen, C.-S.

Chen, W.-C.

Chen, Z.

Chenault, D. B.

D. B. Chenault, J. P. Vaden, D. A. Mitchell, and E. D. Demicco, “Infrared polarimetric sensing of oil on water,” Mar. Technol. Soc. J. 52, 13–22 (2018).
[Crossref]

Chipman, R. A.

Compain, E.

Cui, N.

Davis, T.

De Martino, A.

Demicco, E. D.

D. B. Chenault, J. P. Vaden, D. A. Mitchell, and E. D. Demicco, “Infrared polarimetric sensing of oil on water,” Mar. Technol. Soc. J. 52, 13–22 (2018).
[Crossref]

Drevillon, B.

Duan, J.

J. Duan, Q. Fu, C. Mo, Y. Zhu, and D. Liu, “Review of polarization imaging for international military application,” Proc. SPIE 8908, 890813 (2013)
[Crossref]

Edmiston, C.

Elson, D. S.

J. Qi and D. S. Elson, “Mueller polarimetric imaging for surgical and diagnostic applications: a review,” J. Biophoton. 10, 950–982 (2017).
[Crossref]

Fang, S.

Fei, H.

Feng, B.

B. Feng, Z. Shi, H. Liu, L. Liu, Y. Zhao, and J. Zhang, “Polarized-pixel performance model for DoFP polarimeter,” J. Opt. 20, 065703 (2018).
[Crossref]

B. Feng, Z. Shi, H. Liu, Y. Zhao, and J. Liu, “Calibration method for equivalent extinction ratio of polarized pixel in integrated micropolarizer array camera,” Proc. SPIE 10605, 106051B (2017).
[Crossref]

Fu, Q.

J. Duan, Q. Fu, C. Mo, Y. Zhu, and D. Liu, “Review of polarization imaging for international military application,” Proc. SPIE 8908, 890813 (2013)
[Crossref]

Garcia, M.

Garnett, R.

S. B. Powell, R. Garnett, J. Marshall, C. Rizk, and V. Gruev, “Bioinspired polarization vision enables underwater geolocalization,” Sci. Adv. 4, eaao6841 (2018).
[Crossref]

Gayet, B.

Goudail, F.

Gruev, V.

Gu, G.

L. Guo, E. Cao, G. Gu, X. Hu, W. Qian, M. Wan, and R. Zhao, “Target recognition method based on polarization parameters,” Proc. SPIE 10155, 101552N (2016).
[Crossref]

Guo, L.

L. Guo, E. Cao, G. Gu, X. Hu, W. Qian, M. Wan, and R. Zhao, “Target recognition method based on polarization parameters,” Proc. SPIE 10155, 101552N (2016).
[Crossref]

Han, J.

Hsu, W.-L.

Hu, H.

Hu, X.

L. Guo, E. Cao, G. Gu, X. Hu, W. Qian, M. Wan, and R. Zhao, “Target recognition method based on polarization parameters,” Proc. SPIE 10155, 101552N (2016).
[Crossref]

Huang, B.

Hui, B.

Ibn-Elhaj, M.

Jia, P.

R. Liu, D. J. Wang, P. Jia, and X. Che, “Gradient sky scene based nonuniformity correction and local weighted filter based denoising,” Optik 174, 748–756 (2018).
[Crossref]

LaCasse, C.

Li, F.-M.

Liang, R.

Liu, D.

J. Duan, Q. Fu, C. Mo, Y. Zhu, and D. Liu, “Review of polarization imaging for international military application,” Proc. SPIE 8908, 890813 (2013)
[Crossref]

Liu, H.

B. Feng, Z. Shi, H. Liu, L. Liu, Y. Zhao, and J. Zhang, “Polarized-pixel performance model for DoFP polarimeter,” J. Opt. 20, 065703 (2018).
[Crossref]

B. Feng, Z. Shi, H. Liu, Y. Zhao, and J. Liu, “Calibration method for equivalent extinction ratio of polarized pixel in integrated micropolarizer array camera,” Proc. SPIE 10605, 106051B (2017).
[Crossref]

Liu, J.

B. Feng, Z. Shi, H. Liu, Y. Zhao, and J. Liu, “Calibration method for equivalent extinction ratio of polarized pixel in integrated micropolarizer array camera,” Proc. SPIE 10605, 106051B (2017).
[Crossref]

Liu, L.

B. Feng, Z. Shi, H. Liu, L. Liu, Y. Zhao, and J. Zhang, “Polarized-pixel performance model for DoFP polarimeter,” J. Opt. 20, 065703 (2018).
[Crossref]

Liu, R.

R. Liu, D. J. Wang, P. Jia, and X. Che, “Gradient sky scene based nonuniformity correction and local weighted filter based denoising,” Optik 174, 748–756 (2018).
[Crossref]

Liu, T.

Long, K.

Luo, H.

Marinov, R.

Marshall, J.

S. B. Powell, R. Garnett, J. Marshall, C. Rizk, and V. Gruev, “Bioinspired polarization vision enables underwater geolocalization,” Sci. Adv. 4, eaao6841 (2018).
[Crossref]

Mitchell, D. A.

D. B. Chenault, J. P. Vaden, D. A. Mitchell, and E. D. Demicco, “Infrared polarimetric sensing of oil on water,” Mar. Technol. Soc. J. 52, 13–22 (2018).
[Crossref]

Mo, C.

J. Duan, Q. Fu, C. Mo, Y. Zhu, and D. Liu, “Review of polarization imaging for international military application,” Proc. SPIE 8908, 890813 (2013)
[Crossref]

Myhre, G.

Namer, E.

Ninkov, Z.

D. Vorobiev, Z. Ninkov, N. Brock, and R. West, “On-sky performance evaluation and calibration of a polarization-sensitive focal plane array,” Proc. SPIE 9912, 99125X (2016).
[Crossref]

Novikova, T.

Pacheco, S.

Pau, S.

Peinado, A.

Pierangelo, A.

Powell, S. B.

S. B. Powell, R. Garnett, J. Marshall, C. Rizk, and V. Gruev, “Bioinspired polarization vision enables underwater geolocalization,” Sci. Adv. 4, eaao6841 (2018).
[Crossref]

S. B. Powell and V. Gruev, “Calibration methods for division-of-focal-plane polarimeters,” Opt. Express 21, 21039–21055 (2013).
[Crossref]

Qi, J.

J. Qi and D. S. Elson, “Mueller polarimetric imaging for surgical and diagnostic applications: a review,” J. Biophoton. 10, 950–982 (2017).
[Crossref]

Qian, W.

L. Guo, E. Cao, G. Gu, X. Hu, W. Qian, M. Wan, and R. Zhao, “Target recognition method based on polarization parameters,” Proc. SPIE 10155, 101552N (2016).
[Crossref]

Rizk, C.

S. B. Powell, R. Garnett, J. Marshall, C. Rizk, and V. Gruev, “Bioinspired polarization vision enables underwater geolocalization,” Sci. Adv. 4, eaao6841 (2018).
[Crossref]

Roussel, S.

Schechner, Y. Y.

Shi, Z.

B. Feng, Z. Shi, H. Liu, L. Liu, Y. Zhao, and J. Zhang, “Polarized-pixel performance model for DoFP polarimeter,” J. Opt. 20, 065703 (2018).
[Crossref]

B. Feng, Z. Shi, H. Liu, Y. Zhao, and J. Liu, “Calibration method for equivalent extinction ratio of polarized pixel in integrated micropolarizer array camera,” Proc. SPIE 10605, 106051B (2017).
[Crossref]

Shwartz, S.

Sun, H.

Sun, X.

Vaden, J. P.

D. B. Chenault, J. P. Vaden, D. A. Mitchell, and E. D. Demicco, “Infrared polarimetric sensing of oil on water,” Mar. Technol. Soc. J. 52, 13–22 (2018).
[Crossref]

Vail, A.

Validire, P.

Vorobiev, D.

D. Vorobiev, Z. Ninkov, N. Brock, and R. West, “On-sky performance evaluation and calibration of a polarization-sensitive focal plane array,” Proc. SPIE 9912, 99125X (2016).
[Crossref]

Wan, M.

L. Guo, E. Cao, G. Gu, X. Hu, W. Qian, M. Wan, and R. Zhao, “Target recognition method based on polarization parameters,” Proc. SPIE 10155, 101552N (2016).
[Crossref]

Wang, D.

Wang, D. J.

R. Liu, D. J. Wang, P. Jia, and X. Che, “Gradient sky scene based nonuniformity correction and local weighted filter based denoising,” Optik 174, 748–756 (2018).
[Crossref]

Wang, X.

West, R.

D. Vorobiev, Z. Ninkov, N. Brock, and R. West, “On-sky performance evaluation and calibration of a polarization-sensitive focal plane array,” Proc. SPIE 9912, 99125X (2016).
[Crossref]

Xia, X.

Xing, H.

Zhang, J.

Zhang, R.

Zhao, R.

L. Guo, E. Cao, G. Gu, X. Hu, W. Qian, M. Wan, and R. Zhao, “Target recognition method based on polarization parameters,” Proc. SPIE 10155, 101552N (2016).
[Crossref]

Zhao, X.

Zhao, Y.

B. Feng, Z. Shi, H. Liu, L. Liu, Y. Zhao, and J. Zhang, “Polarized-pixel performance model for DoFP polarimeter,” J. Opt. 20, 065703 (2018).
[Crossref]

B. Feng, Z. Shi, H. Liu, Y. Zhao, and J. Liu, “Calibration method for equivalent extinction ratio of polarized pixel in integrated micropolarizer array camera,” Proc. SPIE 10605, 106051B (2017).
[Crossref]

Zhou, W.

Zhu, Y.

J. Duan, Q. Fu, C. Mo, Y. Zhu, and D. Liu, “Review of polarization imaging for international military application,” Proc. SPIE 8908, 890813 (2013)
[Crossref]

Appl. Opt. (6)

J. Biophoton. (1)

J. Qi and D. S. Elson, “Mueller polarimetric imaging for surgical and diagnostic applications: a review,” J. Biophoton. 10, 950–982 (2017).
[Crossref]

J. Opt. (1)

B. Feng, Z. Shi, H. Liu, L. Liu, Y. Zhao, and J. Zhang, “Polarized-pixel performance model for DoFP polarimeter,” J. Opt. 20, 065703 (2018).
[Crossref]

J. Opt. Soc. Am. A (2)

Mar. Technol. Soc. J. (1)

D. B. Chenault, J. P. Vaden, D. A. Mitchell, and E. D. Demicco, “Infrared polarimetric sensing of oil on water,” Mar. Technol. Soc. J. 52, 13–22 (2018).
[Crossref]

Opt. Express (11)

A. Pierangelo, A. Benali, M.-R. Antonelli, T. Novikova, P. Validire, B. Gayet, and A. De Martino, “Ex-vivo characterization of human colon cancer by Mueller polarimetric imaging,” Opt. Express 19, 1582–1593 (2011).
[Crossref]

W.-L. Hsu, G. Myhre, K. Balakrishnan, N. Brock, M. Ibn-Elhaj, and S. Pau, “Full-Stokes imaging polarimeter using an array of elliptical polarizer,” Opt. Express 22, 3063–3074 (2014).
[Crossref]

G. Myhre, W.-L. Hsu, A. Peinado, C. LaCasse, N. Brock, R. A. Chipman, and S. Pau, “Liquid crystal polymer full-Stokes division of focal plane polarimeter,” Opt. Express 20, 27393–27409 (2012).
[Crossref]

J. Zhang, H. Luo, B. Hui, and Z. Chang, “Image interpolation for division of focal plane polarimeters with intensity correlation,” Opt. Express 24, 20799–20807 (2016).
[Crossref]

E. Namer, S. Shwartz, and Y. Y. Schechner, “Skyless polarimetric calibration and visibility enhancement,” Opt. Express 17, 472–493 (2009).
[Crossref]

S. Fang, X. Xia, H. Xing, and C. Chen, “Image dehazing using polarization effects of objects and airlight,” Opt. Express 22, 19523–19537 (2014).
[Crossref]

B. Huang, T. Liu, J. Han, and H. Hu, “Polarimetric target detection under uneven illumination,” Opt. Express 23, 23603–23612 (2015).
[Crossref]

J. Zhang, H. Luo, R. Liang, W. Zhou, B. Hui, and Z. Chang, “PCA-based denoising method for division of focal plane polarimeters,” Opt. Express 25, 2391–2400 (2017).
[Crossref]

S. Roussel, M. Boffety, and F. Goudail, “Polarimetric precision of a micropolarizer grid-based camera in the presence of additive and Poisson shot noise,” Opt. Express 26, 29968–29982 (2018).
[Crossref]

S. B. Powell and V. Gruev, “Calibration methods for division-of-focal-plane polarimeters,” Opt. Express 21, 21039–21055 (2013).
[Crossref]

A. Ahmed, X. Zhao, V. Gruev, J. Zhang, and A. Bermak, “Residual interpolation for division of focal plane polarization image sensors,” Opt. Express 25, 10651–10662 (2017).
[Crossref]

Optica (2)

Optik (1)

R. Liu, D. J. Wang, P. Jia, and X. Che, “Gradient sky scene based nonuniformity correction and local weighted filter based denoising,” Optik 174, 748–756 (2018).
[Crossref]

Proc. SPIE (4)

L. Guo, E. Cao, G. Gu, X. Hu, W. Qian, M. Wan, and R. Zhao, “Target recognition method based on polarization parameters,” Proc. SPIE 10155, 101552N (2016).
[Crossref]

J. Duan, Q. Fu, C. Mo, Y. Zhu, and D. Liu, “Review of polarization imaging for international military application,” Proc. SPIE 8908, 890813 (2013)
[Crossref]

D. Vorobiev, Z. Ninkov, N. Brock, and R. West, “On-sky performance evaluation and calibration of a polarization-sensitive focal plane array,” Proc. SPIE 9912, 99125X (2016).
[Crossref]

B. Feng, Z. Shi, H. Liu, Y. Zhao, and J. Liu, “Calibration method for equivalent extinction ratio of polarized pixel in integrated micropolarizer array camera,” Proc. SPIE 10605, 106051B (2017).
[Crossref]

Sci. Adv. (1)

S. B. Powell, R. Garnett, J. Marshall, C. Rizk, and V. Gruev, “Bioinspired polarization vision enables underwater geolocalization,” Sci. Adv. 4, eaao6841 (2018).
[Crossref]

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Figures (13)

Fig. 1.
Fig. 1. Signal transmission process of the photoelectric imaging system. A/D, analog-to-digital.
Fig. 2.
Fig. 2. Structure of the polarization detector.
Fig. 3.
Fig. 3. Polarized transmission direction of linearly polarized light and polarized pixels.
Fig. 4.
Fig. 4. Conversion diagram of optical radiation signal to electrical signal.
Fig. 5.
Fig. 5. Optical diagram of our experimental setup.
Fig. 6.
Fig. 6. Constructed experimental setup.
Fig. 7.
Fig. 7. Response of a super-pixel under reference conditions.
Fig. 8.
Fig. 8. Comparison between calculated values and measured values under different exposure times and different linearly polarized light.
Fig. 9.
Fig. 9. Comparison between calculated values and measured values under different exposure times and given linearly polarized light.
Fig. 10.
Fig. 10. Comparison between calculated values and measured values under different gains and different linearly polarized light.
Fig. 11.
Fig. 11. Comparison between calculated values and measured values under different gains and given linearly polarized light.
Fig. 12.
Fig. 12. Comparison between calculated values and measured values under different gamma values and different linearly polarized light.
Fig. 13.
Fig. 13. Comparison between calculated values and measured values under different gamma values and given linearly polarized light.

Tables (7)

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Table 1. Experimental Setup Model

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Table 2. RMSE Values between Calculated Results and Measured Results at Different Exposure Times

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Table 3. RMSE Values between Calculated Results and Measured Results at Different Gains

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Table 4. RMSE Values between Calculated Results and Measured Results at Different Gamma Values

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Table 5. Average RMSE of Polarized Pixels for Each Polarization Orientation at Different Exposure Times

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Table 6. Average RMSE of Polarized Pixels for Each Polarization Orientation at Different Gains

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Table 7. Average RMSE of Polarized Pixels for Each Polarization Orientation at Different Gamma Values

Equations (25)

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S = c r γ ,
S = c ( r + ε ) γ .
L o u t = μ ( i , j ) I cos ( φ z θ ( i , j ) ) 2 + μ ( i , j ) I sin ( φ z θ ( i , j ) ) 2 ,
d ( i , j , z ) = g L o u t + b ( i , j ) ,
d ( i , j , z ) = β g β t g L o u t + b ( i , j ) .
d ( i , j , z ) = β g β t { g [ μ ( i , j ) I cos ( φ z θ ( i , j ) ) 2 + μ ( i , j ) I sin ( φ z θ ( i , j ) ) 2 ] } + b ( i , j ) .
η ( i , j ) = g μ ( i , j ) ,
η ( i , j ) = g μ ( i , j ) .
d ( i , j , z ) = β g β t [ η ( i , j ) I cos ( φ z θ ( i , j ) ) 2 + η ( i , j ) I sin ( φ z θ ( i , j ) ) 2 ] + b ( i , j ) ,
S ( i , j , z ) = { β g β t [ η ( i , j ) I cos ( φ z θ ( i , j ) ) 2 + η ( i , j ) I sin ( φ z θ ( i , j ) ) 2 ] + b ( i , j ) } γ .
cos ( φ z θ ( i , j ) ) 2 = 1 + cos [ 2 ( φ z θ ( i , j ) ) ] 2 ,
sin ( φ z θ ( i , j ) ) 2 = 1 cos [ 2 ( φ z θ ( i , j ) ) ] 2 .
S ( i , j , z ) = { 1 2 β g β t I [ ( η ( i , j ) + η ( i , j ) ) + ( η ( i , j ) η ( i , j ) ) × cos ( 2 φ z 2 θ ( i , j ) ) [ ( η ( i , j ) + η ( i , j ) ) + ( η ( i , j ) η ( i , j ) ) ] + b ( i , j ) } γ ,
η ( i , j ) + = 1 2 ( η ( i , j ) + η ( i , j ) ) ,
η ( i , j ) = 1 2 ( η ( i , j ) η ( i , j ) ) ,
S ( i , j , z ) = { [ β g β t I ( η ( i , j ) + + η ( i , j ) cos 2 φ z cos 2 θ ( i , j ) + η ( i , j ) sin 2 φ z sin 2 θ ( i , j ) ) ] + b ( i , j ) } γ ,
[ S ( i , j , z ) 1 S ( i , j , z ) 2 S ( i , j , z ) n ] [ b ( i , j ) 1 b ( i , j ) 2 b ( i , j ) n ] = I [ 1 1 1 cos 2 φ 1 cos 2 φ 2 cos 2 φ n sin 2 φ 1 sin 2 φ 2 sin 2 φ n ] [ η ( i , j ) + η ( i , j ) cos 2 θ ( i , j ) η ( i , j ) sin 2 θ ( i , j ) ] .
[ y ( i , j ) 1 y ( i , j ) 2 y ( i , j ) n ] = [ S ( i , j , z ) 1 S ( i , j , z ) 2 S ( i , j , z ) n ] [ b ( i , j ) 1 b ( i , j ) 2 b ( i . j ) n ] ,
[ q 1 q 2 q 3 ] = [ η ( i , j ) + η ( i , j ) cos 2 θ ( i , j ) η ( i , j ) sin 2 θ ( i , j ) ] .
[ q 1 q 2 q 3 ] = 1 I [ 1 1 1 cos 2 φ 1 cos 2 φ 2 cos 2 φ n sin 2 φ 1 sin 2 φ 2 sin 2 φ n ] [ y ( i , j ) 1 y ( i , j ) 2 y ( i , j ) n ] .
θ ( i , j ) = 1 2 arctan ( q 3 q 2 ) ,
η ( i , j ) = q 1 2 + q 2 2 cos 2 θ ( i , j ) ,
η ( i , j ) = q 1 2 q 2 2 cos 2 θ ( i , j ) ,
R M S E = 1 n i = 1 n [ y i ϕ ( x i ) ] 2 ,
20 log ( y ) = ( x ) d B ,

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