The architecture of imaging polarimeters with front-mounted polarizer generally used in infrared imaging polarimetry has significant influence on the imaging process of the systems and further calculation of the polarization information of the observed scenario. In this study, the imaging process of infrared polarization imaging system with front-mounted polarizer is analyzed, a radiation correction method based on the modified infrared imaging model of such a system is proposed, and both laboratory and outdoor experiments are performed to verify its effect. Experimental results show that the proposed correction method can effectively eliminate the adverse effects of the radiation introduced by front-mounted polarizer, which significantly reduces scene radiation measurement error and improves the calculation accuracy of the polarization information.
© 2016 Optical Society of America
Polarization is an important characteristic of electromagnetic radiation. Experiments show that the radiation reflected and emitted by objects carries particular polarization information determined by the properties of the objects, such as roughness, moisture, and material, providing useful reference information for detection. With the development of infrared focal plane array (IRFPA) technology, sensitivity of infrared detectors has been improved significantly. This contributes to the application of infrared polarization imaging technology and improves the probability of object detection under complex or strong interference background [1–3].
According to the acquisition modes of polarized images, polarization imaging systems can be divided into division of time imaging polarimeter and simultaneous measurement imaging polarimeter. The former acquires different polarized images at different times and is suitable for observing a static scenario. In such systems, a polarizer or a polarizing beam splitter is often used as a polarization analyzer. Unlike the division of time imaging polarimeter, simultaneous measurement imaging polarimeter is able to obtain four different polarized images through single exposure and is suitable for both stationary and moving objects. Division of amplitude, division of aperture, and division of focal plane are three typical imaging polarimetry modes of such systems . The architecture of polarimeters with a polarizer mounted in front of the optical system or imaging focal plane, which is simple and inexpensive, is generally used in imaging polarimetry. Examples of infrared polarization imaging systems [5–7] with front-mounted polarizers are shown in Fig. 1.
In visible polarization imaging systems, the front-mounted polarizer basically will not introduce any additional visible light that does not arise from the observed scenario. However, in infrared imaging polarimeters with front-mounted polarizer, infrared radiation from both the observed scenario and front-mounted polarizer will be received by IRFPA. This will lead to distortion in the incident radiation, thus affecting further calculation of the polarization information of the observed scenario. Particularly in uncooled long-wave infrared (LWIR) polarization imaging systems, the distortion caused by the front-mounted polarizer is more severe. To solve this problem, we first theoretically analyze the imaging process of the infrared polarization imaging system with the front-mounted polarizer, then present a radiation correction method based on the analysis, and finally validate the effectiveness of the method via two experiments.
2. Mathematical model of infrared polarization imaging system with front-mounted polarizer
A. Principle of imaging polarimetry based on the Stokes vector
The Stokes vector [8,9] is a 4 × 1 matrix, which is defined as
In addition, degree of linear polarization (DoLP) and angle of polarization (AoP) are also used to describe the polarization characteristics of radiation. The former defines the fraction of the intensity attributed to linear polarized radiation states while the latter defines the angle of orientation in which the polarized energy is the strongest with respect to x-axis. Mathematically, in terms of the Stokes-vector elements,
The effect of optical component or system on incident Stokes vector can be described using a 4 × 4 Mueller matrix, which is defined as
Then, the Stokes vector of incident radiation Si = [Ii Qi Ui Vi]T is transformed into the vector of exit radiation So = [Io Qo Uo Vo]T by
Thus, the radiation intensity received by the imaging device is
By changing the state of polarization of the imaging system four times, we obtain four equations similar to Eq. (5). If any two of them are independent of each other, the Stokes parameters can be calculated from the four intensity equations. Generally, a circularly polarized component is assumed to be considerably weak in natural scenarios. Therefore, the V component is usually ignored. Thus, all the Stokes parameters, except V, can be calculated from three intensity equations.
B. Ideal model of the infrared polarization imaging system with front-mounted polarizer
Figure 2 shows the schematic diagram of an ideal imaging process of an infrared polarization imaging system with a front-mounted wire grid polarizer. The incident radiation from the scenarios pass through the polarizer and object lens successively and finally arrive at the image plane. Images of different polarization channels can be captured with the polarizer’s pass axis rotating at different angles, while the image of intensity channel can be captured directly without the polarizer.
According to theoretical derivation [8,10], the Mueller matrix of the wire grid polarizer can be denoted as
Assuming that the object lens causes no energy loss, we can obtain a relationship between the radiation intensity received by a pixel and the Stokes parameters of the incident radiation from Eq. (5) and Eq. (6):
Denoting the three measurement angles of the polarizer’s pass axis with respect to the x axis as θi (i = 0, 1, 2), and defining a 3 × 1 intensity column vector I = [I0 I1 I2]T, in which Ii (i = 0, 1, 2) is the intensity measured at orientation θi, we can describe the measurement process as follows 
Denoting the radiation intensity received by a certain pixel at orientation α as Iα and at its orthogonal orientation α + π/2 as Iα + π/2, we can obtain the following equations from Eq. (7):
Then, the relationship among Iα, Iα+π/2, and INP can be derived as
C. Modified model of the infrared polarization imaging system with front-mounted polarizer
The ideal model mentioned in Sec. 2. A is established with several factors omitted and is thus different from the actual model. The main defects of the ideal model are as follows:
- 1. The transmission of the wire grid polarizer at its pass orientation and orthogonal orientation is not considered.
- 2. The energy loss caused by the object lens is not considered.
- 3. The infrared radiation reflected and emitted by the polarizer, which increases the intensity received by the image plane, is not considered.
Here, the modified model of the infrared polarization imaging system with front-mounted polarizer is given based on the ideal model.
As shown in Fig. 3, Sin is the linear Stokes vector of the incident radiation from the scenario, and R is the additional intensity caused by the reflected and emitted radiations from the polarizer. 𝜏1 and 𝜏2 represent the transmission of the wire grid polarizer at its pass orientation and orthogonal orientation, respectively, while ρ represents the energy transmission of the object lens. By taking the transmission of the polarizer into account, the coefficient matrix Mc becomes 
Thus, the radiation intensity received by the pixel is
Then, Iα and Iα+π/2 are given asEq. (12) no longer holds true in the modified model.
3. Radiation correction method based on the modified infrared polarization imaging system with front-mounted polarizer
Rotating the wire grid polarizer and measuring the intensity received at the corresponding polarization orientations for three times, we obtain the following equation according to the modified model given in Sec. 2. C,
By measuring the radiation intensity again, but with the polarizer removed, we obtain
In the condition that 𝜏1, 𝜏2, and ρ are already known, the linear Stokes parameters and additional intensity r at each pixel can be calculated according to Eq. (20) and the radiation intensity equations. In this manner, we can eliminate the adverse effects of the polarizer and restore the correct polarization information of the observed scenario.
4. Calibration of the mapping relationship between image grayscale and irradiance
Because the previous analysis is based on the radiation intensity rather than image grayscale, we need to determine the conversion relationship between image grayscale and the radiation intensity received by the detector. In general, there is a nonlinear mapping relationship between the detector output signal and the intensity of the incident radiation , which is described as follows
Figure 4 shows the LWIR polarization imaging system used to test the radiation correction method in this study. It consists of a rotating front-mounted polarizer and a LWIR thermal camera; the former is detachable such that the radiation intensity can be captured directly. The technical specifications of these two components are shown in Table 1 and Table 2.
The calibration method adopted in this study is similar to that mentioned in the third chapter of Ref , and the experimental setup is shown in Fig. 5. Using the IR-2100 extended area blackbody source from Infrared Systems Development Corp., we accomplished the calibration of the LWIR camera (without polarizer) and obtained the fit function in the form of Eq. (21) between the average grayscale of the thermal image of the blackbody and radiant exitance within 7.5–13.5 μm. The emissivity of the blackbody source is 0.96, and its temperature resolution is 0.1 K. In the calibration experiment, we set the temperature range of the blackbody as 278.15–393.15 K. Within this temperature range, we sampled a temperature value and captured an image of the blackbody at every 5 K intervals. The radiant exitance of the blackbody within 7.5–13.5 μm can be calculated according to the Planck’s law of blackbody radiation. Because the distance between the blackbody and imaging system is longer than ten times the latter’s focal length, the irradiance at the image plane is approximately proportional to the radiance of the blackbody source. On the other hand, the radiance of the blackbody is proportional to its radiant exitance . Therefore, the irradiance at the image plane of the detector is also approximately proportional to radiant exitance of the blackbody. This implies that the radiant exitance can be regarded as “relative irradiance” at image plane of the detector. By the nonlinear least squares fitting, a, γ, and b are determined to be 24.03, 0.9741, and −123.7, respectively (RMSE = 16.86). Thus, the “relative irradiance” can be calculated according to the image grayscale and the mapping function. The fitting curve of the image grayscale and “relative irradiance” is shown in Fig. 6.
5. Validation experiments and analysis of the effectiveness of the radiation correction method
A. Laboratory experiment and analysis of the radiation Correction of linearly polarized blackbody radiation source
A blackbody is assumed to be unpolarized [15, 16]; thus, blackbody radiation should be completely linearly polarized after passing through a wire grid polarizer. Based on such a premise, we designed a linearly polarized blackbody radiation source, the DoLP and AoP of which are already known. Then, the steps to test the effectiveness of proposed radiation correction method are as follows
- 1. Capture the polarized images and intensity image of the linearly polarized blackbody radiation source using the LWIR polarization imaging system shown in Fig. 4.
- 2. Convert the grayscale of the images to “relative irradiance” according to the mapping function mentioned in Sec. 4.
As shown in Fig. 7, there is a wire grid polarizer in front of the IR-2100 extended area blackbody source, which acts as the polarization state generator (PSG). The pass axis of the polarizer is 45° with respect to x-axis. In the experiment, the temperature of the blackbody was set as 323.15 K, 333.15 K, 343.15 K, 353.15 K, 363.15 K, and 373.15 K, successively. At each temperature, the polarized images of 45°, 90°, and 135° polarization channels and the intensity image are captured. Following step (2) and (3), we obtain the uncorrected and corrected average results of the linear Stokes parameters, DoLP, and AoP at different blackbody temperatures, which are shown in Table 3. Here, we are only concerned with the relative relationship among the Stokes parameters and assume that ρ = 1.
From Table 3, it can be seen that
(1) Both uncorrected results and corrected results of the linear Stokes parameters I, Q, U, and DoLP increase with a rise blackbody temperature. However, the calculated result of DoLP is much smaller than its ideal value, which should be 1.
Actually, the infrared radiation received by polarization imaging system consists of not only the transmitted linearly polarized radiation from the polarizer in front of the blackbody, but also the infrared radiation emitted and reflected by the polarizer, the latter of which cannot be omitted. With increasing blackbody temperature, the intensity of the radiation from the blackbody increases. Thus, the transmitted linearly polarized radiation from the polarizer becomes stronger, and its proportion to the total radiation received by the imaging system increases. This is the reason why the calculated DoLP increases with the increase in the temperature of blackbody regardless of whether it is corrected or not.
For the polarization imaging system shown in Fig. 7 under room temperature, which is 298.15 K, its imaging object is actually the polarization state generator in front of the blackbody. Assuming that the polarizer is a gray body and the additional infrared radiation, except the transmitted linearly polarized part, is simply emitted by the polarizer, we can estimate the DoLP of polarized blackbody radiation source in Fig. 7 asEq. (22), which are shown in Table 4.
From Table 4, it can be seen that the infrared radiation coming from the polarized blackbody radiation source is not completely linearly polarized. With decreasing εP or increasing TBB, DoLPPBRS increases. Moreover, under the condition that only the radiation emitted by the polarizer is considered, the original DoLP of the polarized blackbody radiation source is already significantly smaller than its ideal value, which is 1. If the reflected radiation is taken into account, there may be a further drop in DoLPPBRS from the current estimated value.
(2) At a certain blackbody temperature, I diminishes, Q, U, and DoLP increase, while the AoP does not change after radiation correction. The calculated result of the AoP is close to the theoretical value 45°, although there still exists some error between them.
The corrected results, uncorrected results, and theoretical estimates of the DoLP of the polarized blackbody radiation source at different temperatures are shown in Fig. 8. As the radiation introduced by the polarization analyzer is eliminated after radiation correction, the total radiant energy received by the detector is lower than that without correction, leading to an improvement in the proportion of the linearly polarized radiation from the polarized blackbody radiation source in the total radiation received. Therefore, the corrected result of the DoLP is much higher than the uncorrected result, thus being closer to the theoretical estimate of the DoLP.
Moreover, the reason why the calculated AoP does not change after correction may be as follows: the radiation introduced by the polarization analyzer is basically unpolarized, and its energy contribution at any orientation is approximately the same. Thus, the orientation at which the polarized energy is the highest does not change, thereby not affecting the calculation result of the AoP. The deviation between the calculated AoP and the theoretical value of 45° may be caused by the manual setup of the pass orientation of the PSG and polarization state analyzer (PSA) in the experiment.
B. Experiment and analysis of the Radiation correction of actual outdoor scenario
Images of an outdoor scenario were captured at approximately 8 pm on a calm summer evening. As shown in Fig. 9, it mainly consists of a car and some buildings in the scene.
In experiment, four groups of images from 45°, 90°, and 135° polarization channels and intensity channel were captured, each of which consisted of 50 frames. To avoid the influence of random noise, we used the average result of 50 frames. Thus, we obtained four average images. To facilitate the observation and comparison, we linearly compressed the original 14-bit-width image data to the same 8-bit-width range, and then obtained the four images, as shown in Fig. 10. It can be seen that the brightness of the polarized images is much higher than that of the intensity image, while the contrast is much lower. This phenomenon results from the radiation introduced by the front-mounted polarizer of the imaging system.
According to the mapping relationship mentioned in Sec. 4, the uncorrected and corrected results of I, Q, U, DoLP, and AoP were calculated using Eq. (9) and Eq. (20), respectively. Then, we obtained five groups of images, as shown in Fig. 11–15. To statistically analyze the information of these images, we generated five groups of histograms, as shown in Fig. 16.
- (1) Compared with uncorrected I’s histogram, the centroid of the directly captured I’s histogram shifts to the left and the shape basically remains unchanged. The brightness of the directly captured I’s image is much lower than that of the uncorrected I’s image.
- (2) After correction, both the centroid location and shape of Q’s histogram basically remain unchanged while the centroid of the U’s histogram slightly shifts to the right. The visual differences between the uncorrected and corrected images of Q, and uncorrected and corrected images of U, are not evident.
- (3) After correction, the centroid location of the DoLP’s histogram shifts to the right, and the shape of histogram widens. The brightness of DoLP’s image increases, particularly in relatively smooth areas such as distant metal roof and rear window of the car; the contrast also increases to a certain extent. The centroid location and shape of the AoP’s histogram basically remain unchanged, and the visual difference between uncorrected and corrected images of the AoP is not evident.
From Fig. 16(d), it can be seen that the DoLP of the scenario is so low that it essentially varies in the range 0–0.04. Then, we can assume that the total energy transmission of the polarizer is equal to the sum of the transmission at its pass orientation and orthogonal orientation. Denoting the radiation intensity received from the polarization channel by the detector as Ip (p = 0, 1, 2) and the ideal intensity of incident radiation from the scenario as (p = 0, 1, 2), we obtain the following approximate relationship
Replacing Ip in Eq. (9) with in Eq. (23), we calculate the Stokes parameter I using Eq. (9) again. The calculation result and directly captured intensity (INP) are shown in Fig. 17. From these two images, it can be seen that there is no obvious difference in the brightness between them, indicating that the additional intensity introduced by the polarizer has already been eliminated successfully.
For an infrared polarization imaging system with a front-mounted polarizer, the intensity of additional radiation introduced by the polarizer can be significantly strong. The calculation result in Sec. 5. B indicates that the additional “relative irradiance” caused by the polarizer reached a value of approximately 130, which is similar to that caused by the actual radiation from the scenario. This will lead to considerable errors in the measurement of radiant intensity and further calculation of the DoLP. Therefore, it is necessary to eliminate the adverse effect of the polarizer.
This study analyzed the imaging process of infrared polarization imaging system with a front-mounted polarizer, a radiation correction method based on the modified infrared imaging model of such a system was proposed, and two experiments, radiation correction experiments of linearly polarized blackbody radiation source and outdoor scene, were performed to validate the effectiveness of the method. Experimental results showed that the proposed correction method can eliminate the adverse effect resulting from the radiation emitted and reflected by the polarizer, which significantly reduces the scene radiation measurement error and improves the calculation accuracy of the polarization information. Because the information of both the polarization channels and intensity channel is required, the proposed radiation correction method is more suitable for a multi-channel imaging system as the front-mounted polarizers do not need adjusting or dismounting.
The Key Program of National Natural Science Foundation of China (Grant No. 61231014); The General Program of National Natural Science Foundation of China (Grant No. 61575023); The Key Pre-study Foundation of General Armament Department of China (9140A01010113BQ01001).
The authors would like to thank Dr. Zhang Xu and Dr. He Si for their help in the calibration experiment.
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