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Abstract
The neutral points are one of the most significant characteristics of the polarized skylight pattern in the whole sky. At present, detection of the neutral points mostly utilizes ellipse fitting of the degree of polarization. However, because the degree of polarization distribution characteristics of a polarized skylight pattern is easily affected by the environment, the robustness of the detection is unstable. Aiming at the problem, we analyzed the angle of polarization distribution characteristics of polarized skylight patterns in the region around the neutral point by measurement experiments. Based on this, we proposed an automatic detection method of neutral points using the angle of polarization of the polarized skylight pattern. The experimental results of different times in a continuous period of time show that compared with ellipse fitting of the degree of polarization, the detection accuracy of the proposed method is almost the same, but the robustness is better. It provides a novel method for the position detecting of the neutral point, which is in favor of the measurement applications of polarization technology.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
The sunlight itself is the natural light without polarization. During the transmission process, because the sunlight is affected by the absorption and scattering of the gas molecules, aerosol particles and cloud particles in the earth's atmosphere and the reflection of the earth surface, it will be transformed the polarized light with partial polarization state. In addition, it presents a stable polarization distribution pattern called “polarized skylight pattern” [1,2] in the whole sky, which is one of the important natural properties of the earth. Biological studies have shown that Cataglyphis ants [3,4], Mouse-eared bat [5], Mantis shrimp [6] and other organisms can use polarized light to obtain location information. Biological polarization perception ability makes researchers have a strong interest in polarized skylight patterns. More and more scholars carry out research on the physical characteristics of polarized skylight patterns [7,8]. The application of the characteristics of polarized skylight patterns has been widely used in navigation, meteorological monitoring and polarization remote sensing [9–11].
The neutral point is one of the most significant characteristics of the polarized skylight pattern. Usually, we can only observe the Babinet neutral point and Arago neutral point or Babinet neutral point and Brewster neutral point at the same time on a clear day [12]. The neutral point position is closely related to atmospheric turbidity, so the observation of the neutral point can guide the monitoring of atmospheric environmental quality [13]. Moreover, degree of polarization (DOP) represents the ratio of the intensity of the polarized part of the beam to the intensity of the whole beam. Due to the extremely low DOP of the region around neutral point, remote sensing detection through the region around the neutral point can reduce the negative impact of atmospheric polarization characteristics on remote sensing imaging, which can obtain more accurate target characteristics [14,15]. The position of neutral point is closely related to the solar position and has obvious orientations, which can provide accurate location information for bionic polarized light navigation [16]. Therefore, detection of the neutral point becomes one of the most important foundations for the subsequent research. Because neutral point is a concept closely related to the DOP of a polarized skylight pattern, the current detection method of the neutral point is mostly based on the characteristics of DOP [17].
However, because the distribution of DOP of polarized skylight patterns is very sensitive to environmental changes [18,19], the detection accuracy of neutral point is greatly affected by environmental factors. Therefore, it is of great significance to explore a more accurate method for detection of neutral point. The angle of polarization (AOP) is defined as the angle between the E-vector direction of the measured point and the local meridian at the point. There is also the case where DOP and the angle of polarization (AOP) are utilized in combination in neutral point detection [20]. In the automated method of neutral point detection, the locations are determined by the condition that the AOP change is maximized while the DOP is minimized.
The AOP of a polarized skylight pattern also contains the important skylight polarization information. Previous studies have shown that AOP is less sensitive to environmental change than DOP [21]. Through observation experiments, DOP and AOP distribution features of the polarized skylight pattern in the whole sky are compared and analyzed. We found that the neutral point is an intersection point of the four regions with AOP of −90°, −45°, 45° and 90°, which is less affected by the environment. On the basis of the analysis results, we proposed a method of neutral point detection by utilizing the AOP of a polarized skylight pattern. In addition, the effectiveness and robustness of the proposed method are verified through automatic detection of the neutral point in different times.
2. Analysis of the neutral point in the AOP image of a polarized skylight pattern
Neutral points, also known as “polarization defects”, are depolarizing effects caused by multiple scattering of aerosol particles and reflected light from the earth surface [22]. In the case of a clear sky, the four neutral points are located on the main plane perpendicular to the solar and the zenith, and their distribution is shown in Fig. 1. The Arago neutral point is a polarization singularity in the sky which was firstly discovered by French astronomer Arago in 1809. Thirty years later, French meteorologist Jacques Babinet and Scottish physicist Sir David Brewster discovered the Babinet neutral point and Brewster neutral point. According to the symmetry characteristics of the neutral point distribution, scientists predicted that there should be the fourth neutral point at the anti-solar position. It was not until 2002 that Gabor Horvath first observed the fourth neutral point by hot air balloon [23].
Fig. 1. Distribution of neutral points on the main plane of the sky. The Babinet is above the sun and Brewster neutral point is below the sun. The Arago is above the anti-sun and the 4th neutral point is below the anti-sun.
The neutral point is the comprehensive result of sunlight scattered by the small particles in the atmosphere. The DOP of the region near the neutral points has the following characteristics:
- (1) The DOP at the neutral point is 0.
- (2) The DOP increases first and then decreases in the region between the two neutral points.
The site of the automatic detection experiments of the neutral point was the East Playground Center of Feicui Lake Campus, Hefei University of Technology (31° 46’ 42” N, 117° 12’ 51” E). The experimental time was 17:30 on January 22, 2019. Figure 2 are the distribution images of polarized skylight pattern in clear day. On a clear day, the variation trend of DOP can be clearly observed in the DOP image. It can be seen from 2(b) and 2(c) in Fig. 2 that there are two regions with extremely low DOP in the image, namely the neutral region, and the DOP increases gradually and then decreases between the two neutral points. According to the distribution characteristics of DOP, scholars proposed a method of DOP elliptic fitting for neutral point detection. In addition, the experiments were carried out under the different clear weather, and the positions of neutral points were obtained more accurately [24].
Studies have shown that DOP of polarized skylight pattern is more sensitive to environmental changes. It is well known that the DOP of polarized skylight pattern in the whole sky is low under the influence of occlusions. The experiments were conducted at the East Playground Center of Feicui Lake Campus, Hefei University of Technology (31° 46’ 42” N, 117° 12’ 51” E). The experimental time was 16:00 on November 14, 2019. Figure 3 shows the distribution images of polarized skylight pattern under the influence of occlusion. The DOP of the regions shaded by trees and houses is low, which is almost close to the DOP of the neutral region. If the method of DOP elliptic fitting is directly used to calculate the position of the neutral point, it is bound to be affected by these invalid regions. However, the distribution characteristics of AOP of polarized skylight pattern have stronger anti-interference performance.
Based on the above analysis, the theoretical basis of position detection of neutral point in the AOP image of polarized skylight pattern will be firstly discussed. According to Stokes vector, the equation of DOP calculation is as follows:
According to the distribution characteristics of neutral point, the neutral point (NP) is defined as:
Combined with Eq. (2), it can be derived that the neutral point satisfies the following relation in the AOP image of a polarized skylight pattern:
- (1) ${U_j}$ is the higher order infinitesimal of ${Q_j}$ if the speed of ${U_j}$ converging to 0 is faster than that of ${Q_j}$ converging to 0. Then Eq. (2) can be simplified to $N{P_{DOP}} = \frac{{{Q_i}}}{{{I_i}}}$, which represents the neutral point line of $Q = 0$. The limit of ${U_j}$ divided by ${Q_j}$ is positive zero or negative zero, which represents $N{P_{AOP}}$ is +90° or −90°. Accordingly, the pixel points whose AOP is ±90° are on the neutral line of $Q = 0$.
- (2) ${U_j}$ is the lower order infinitesimal of ${Q_j}$ if the speed of ${U_j}$ converging to 0 is slower than that of ${Q_j}$ converging to 0. Then Eq. (2) can be simplified to $N{P_{DOP}} = \frac{{{U_i}}}{{I_i^{}}}$, which represents the neutral point line of $U = 0$. Moreover, the limit of ${U_j}$ divided by ${Q_j}$ is positive ∞ or negative ∞, which represents $N{P_{AOP}}$ is +45° or −45°. Accordingly, the pixel points whose AOP is ±45° are on the neutral line of $U = 0$.
- (3) ${U_j}$ is the same order infinitesimal of ${Q_j}$ if the speed of ${U_j}$ converging to 0 is equivalent that of Eq. (3) ${Q_j}$ converging to 0. The limit of ${U_j}$ divided by ${Q_j}$ is c, which the AOP of the region of neutral point are non ±90° or ±45° in this case. It shows that the neutral point is not an ideal point and it can be easily disturbed by other factors resulting in noise.
According to the above analysis about neutral point model based on the AOP of polarized skylight pattern, the neutral point is an intersection point of the four regions with AOP of −90°, −45°, 45° and 90°. Therefore, in this paper, the AOP of polarized skylight pattern is quantified, and the image is quantified into four regions including −90° region, −45° region, 45° region and 90° region, which can reduce data redundancy. The distribution of neutral points in the AOP image is shown more intuitively. The experimental results are shown in Fig. 4.
3. Neutral point detection method
The value range of AOP images is between $- \pi /2$ and $\pi /2$. We further process AOP images for the convenience of subsequent calculations.
According to the results of the second section, a method of neutral point detection by calculating the local entropy (LE) of the AOP image is proposed. Image entropy is a statistical form of features, which reflects the average amount of information in an image [25,26]. In order to characterize the spatial characteristics of gray value distribution of image, the two-dimensional entropy of image is introduced. The two-dimensional entropy of image can further reflect the gray information of a pixel.
Firstly, the neighborhood gray mean of the image is selected as the spatial characteristic quantity of gray distribution. Combined with the pixel gray of the image, a binary combination of characteristic is formed, which is recorded as $(i,j)$. Where i represents the gray value of the pixel $(0 \le i \le 255)$, and j represents the gray mean of the neighborhood $(0 \le j \le 255)$:
where $f(i,j)$ is the frequency of the binary combination of characteristic $(i,j)$, and N is the scale of the image. Therefore, the two-dimensional entropy of the image is:The LE image of the AOP image for polarized skylight pattern is calculated at one time shown as Fig. 5. It can be seen from the figure that the intersection of the curves composed of discrete points is the neutral point. And according to the property of LE, the entropy value at the neutral point is the largest. If the positions of the point with the maximum entropy in the image are directly taken as the neutral point, the experiment results are easily affected by the error points. A suitable threshold is found through the experiment, and then ellipse fitting is carried out for the value region. The center position of the fitting ellipse is taken as the neutral point, which can reduce the influence of very few points in the image on the experimental results.
A large number of observation experiments show that the LE in the region of non-neutral point is extremely low, but the LE in the region of neutral point is greater than 1. Therefore, the LE image is binary processed by setting threshold of 1. Then the mathematical morphological closure is utilized to smooth the edges of binary images. Next, canny operator is adopted to detect the edge region of the smooth binary image. Finally, the least square method is used to fit the processed edge curve [27]. The fitted ellipse can be expressed by quadratic polynomial:
where ${\boldsymbol a} = [a,b,c,d,e,f]$ and ${\boldsymbol u} = {[{x^2},xy,{y^2},x,y,1]^T}$. By solving the generalized eigenvalues of Eq. (8), the least squares elliptic fitting equation can be obtained. The center $C({x_c},{y_c})$ of the ellipse is the neutral point:In conclusion, Fig. 6 shows the detection process of the neutral point:
According to the above algorithm steps, the detection process of neutral point is demonstrated in detail by using a set of measured data, the processing results of stepwise procedure is shown in Fig. 7. Firstly, three polarization images with different angles are gained by the all-sky imaging device of polarized skylight pattern and AOP image can be obtained by solving Stokes equations, as shown in 7(a) and 7(b) of Fig. 7. Then the AOP image is quantized and the result is as shown in Fig. 7(c). According to Eq. (6) and Eq. (7), the local entropy (LE) image is obtained, and the binary image can be achieved by using threshold value for LE image, whose results are shown in Figs. 7(d) and 7(e). The image edge is smoothed by closing operation of morphology and detected by canny operator, as shown in Fig. 7(f). Finally, the least square method is utilized to fit the edge of the image, and the center of the fitting is the position of the neutral point. The final detection results are shown in Fig. 7(g).
4. Experiments and discussion
4.1 Detection of polarized skylight pattern
All-sky imaging device of polarized skylight pattern is mainly composed of Sigma 8 mm F/3.5 fisheyes lens, Thorlabs relay lens, polarizer, macro lens and three-channel polarization measurement system [28], as shown in Fig. 8. The field angle of fisheye lens is 142°, which can capture the light intensity information of the whole sky. The rotation polarizer is controlled by a single chip microcomputer to obtain the information of sunlight intensity from different angles. The three-channel polarization measurement system realizes the perception and recording of sunlight intensity information. In order to obtain accurate information of AOP, it is necessary to calibrate the all-sky imaging device of polarized skylight pattern from three aspects: the correction coefficient of the polarization direction, gray response of the polarizer and the pixel offset of the three-channel polarization camera [29].
The main experimental steps were as follows: (1) Adjusted the equipment to maintain the level so that the imaging center was facing the zenith and the main axis of the lens was pointing to the geographic north; (2) Rotated the 0° polarizer to made its optical axis parallel to the direction of geographic north to obtain the light intensity value of the first polarization image; (3) Rotated the polarizer clockwise by 60° and 120° respectively to obtain the light intensity values of the other two polarization images; (4) Substituted the three polarization images into Eq. (10) of the Stokes vector to calculate the $[I,Q,U]$ of the polarized skylight pattern. And then use Eq. (1) and Eq. (3) to calculate the AOP and DOP of the sky region at the current observed time.
4.2 Experimental results
Since the Arago neutral point is distributed about 30° above the anti-solar position. In order to realize observation of the two neutral points at different times in the continuous period of time, the observation experiment was carried out in the different period with large zenith angle. The site of the automatic detection experiments of the neutral point was the East Playground Center of Feicui Lake Campus, Hefei University of Technology (31° 46’ 42” N, 117° 12’ 51” E). The experimental time was from 08:30 to 17:30 on January 22, 2019. The time interval was 1 hour and the weather was sunny. Firstly, the all-sky imaging device of polarized skylight pattern was used to obtain experimental data at different times. Then, according to Eq. (1) and Eq. (3), the AOP images of polarized skylight pattern was calculated at different times, as shown in Fig. 9.
The positions of Arago neutral point and Babinet neutral point were calculated by the proposed method and compared with the actual position of neutral point. Five colleagues in our team manually marked the position of neutral point in the measured DOP image of the polarized skylight pattern repeatedly and independently. Eventually, the average value of the five results is taken as the actual position of neutral point. The experimental results of neutral point detection at different times were shown in Fig. 10.
In order to describe the error of the neutral point detected by the proposed method, the detecting error k of the neutral point were defined as the ratio of the Euclidean distance and image size, The Euclidean distance is expressed as the linear distance between the calculated pixel position of neutral point $({x_{alculate}},{y_{alculate}})$ and the actual pixel position of neutral point $({x_{actual}},{y_{actual}})$. The image in this paper is 1680*1680 pixels, so N=1680 in equation.
According to Eq. (11), the detecting error of the two neutral points (Arago and Babinet) in different time of Fig. 9 was calculated respectively. At the same time, the error of neutral point position detected by the degree of polarization ellipse fitting (DOP Fit) method [17] is compared with our method, shown in Fig. 11.
Fig. 11. Detecting error curves of the two neutral points (Arago and Babinet) by two methods in clear day.
The maximum detecting errors of Arago neutral point and Babinet neutral point calculated by proposed method were 0.0131 and 0.0214 respectively, and the minimum detecting errors were 0.0065 and 0.0119 respectively. The average detecting error was 0.0089 and 0.0149 respectively. Through comparing the difference between the detecting errors of Arago neutral point and those of the Babinet neutral point in different times, it can be found that the detecting errors of Arago neutral points were smaller than those of Babinet neutral points. It was because that the Babinet neutral point is on the side of the solar position, which was easily affected by direct sunlight. The maximum detecting errors of Arago neutral point and Babinet neutral point calculated by DOP Fit were 0.0208 and 0.0262 respectively, and the minimum detecting errors were 0.0060 and 0.1111 respectively. The average detecting error was 0.0095 and 0.0196 respectively. By comparing the experimental results of different methods, the detection accuracy of Argo neutral point is comparable. But the detection accuracy of Babinet neutral point is better than that of DOP Fit method.
In order to further analyze the robustness of the neutral points calculated by the method in this paper, the detecting variance ${s^2}$ of neutral point is defined as:
where, n is the total number, ${k_i}$ is the detecting error of each neutral point, and $\overline k$ is the average value of detecting error.The two detecting variances of neutral points in different times were calculated by different methods respectively. Detecting variances of Arago and Babinet neutral points are 13.80 and 19.33 by proposed method in this paper respectively. Detecting variances of Arago and Babinet neutral points are 81.40 and 63.18 by DOP Fit method respectively. The detecting variances of the two neutral points calculated by our method were both lower, which indicates that the proposed method had stronger robustness.
Under the condition of tree occlusion, the local information of polarized skylight pattern was destroyed. In order to analyze the influence of tree occlusion on accuracy of the detection of neutral point furtherly, the experiments were also carried out. The experiments were conducted at the East Playground Center of Feicui Lake Campus, Hefei University of Technology (31° 46’ 42” N, 117° 12’ 51” E). The experimental time was from 15:00 to 17:00 with the interval of half an hour on November 14, 2019. The experimental data of polarized skylight pattern in different times in the continuous period of time were collected by the all-sky imaging device of polarized skylight pattern, as shown in Fig. 12.
In the case of occlusion, the polarization information of trees and houses is weaker and the degree of polarization is lower, which is easy to interfere with the position detection of neutral point. In this paper, the global information of AOP image of polarized skylight pattern was used to detect the position of neutral point. On one hand, the influence of noise was reduced by quantifying the AOP images. On the other hand, the two neutral lines in the AOP images can be automatically obtained. The detecting error curves of the two neutral points (Arago and Babinet) under occlusion by different methods were shown in Fig. 13.
Fig. 13. Detecting error curves of the two neutral points (Arago and Babinet) by two methods under occlusion.
According to Fig. 13, the error results present smaller error of the neutral point detection by the proposed method under occlusion. Moreover, the detecting variances of the neutral points detected by our method is smaller than that of DOP Fit method. The robustness of the method based on AOP of polarized skylight pattern is better than DOP Fit method, which is less affected by the occlusion condition.
5. Conclusion
In this paper, the distribution characteristics of polarized skylight pattern are analyzed through a large number of observing experiments. The characteristic relationship between the neutral point in the DOP distribution map and AOP distribution map is further obtained. Based on this, an automatic detection method of the neutral point is proposed by utilizing the AOP image of polarized skylight pattern. The detecting experiments and error analysis on the neutral point of polarized skylight pattern in different time are carried out for verifying the validity and accuracy of the proposed method. The experimental results show that small detecting error of the neutral points detected by the proposed method can be achieved. Especially in the case of intense light with high solar zenith angle, the detecting error and variances of the neutral point remains small. In summary, the method proposed in this paper can achieve a better detection performance with high accuracy and strong robustness, which provides a novel method for application of the neutral point in the field of polarized light navigation, polarization remote sensing and meteorological monitoring.
Funding
National Natural Science Foundation of China (61571177); University Natural Science Research Project of Anhui Province (KJ2018JD12).
Acknowledgments
Zhiguo Fan acknowledges the National Natural Science Foundation of China.
Disclosures
The authors declare no conflicts of interest.
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