﻿ 多云天气条件下的大气偏振光定向方法
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Atmospheric polarized light orientation method in cloudy weather
FAN Ying, HE Xiaofeng, FAN Chen, HU Xiaoping, WU Xuesong, HAN Guoliang, LUO Kaixin
College of Intelligence Science and Technology, National University of Defense Technology, Changsha 410073, China
Abstract: Aiming at the problem that the precision of the traditional polarized light orientation algorithm decreases in cloudy weather, this paper proposes a method for atmospheric polarized light orientation in cloudy weather. First, the directional model of polarized light for any pixel is established according to the known information; then, based on the directional model, the interior points that meet the requirements are screened out using the random sampling consistent algorithm, and the most satisfactory directional model is achieved according to the selected interior points; finally, the optimal directional model is used to calculate the heading angle, thereby realizing the orientation via atmospheric polarized light. The validity of the method is proved by the actual data, and the calculated heading angle errors are smaller than 0.5° and 1° in partly cloudy and cloudy weather, respectively.
Keywords: polarized light    cloudy weather    orientation    random sampling    consensus algorithm

1 大气偏振光定向算法 1.1 大气偏振光定向原理

 图 1 大气偏振光分布示意图 Fig. 1 Diagram of polarized light distribution in atmosphere

 图 2 偏振光定向原理图 Fig. 2 Schematic diagram of polarized light orientation

1.2 大气偏振光定向模型的建立

 $\mathit{\boldsymbol{a}}_p^{\rm{b}} = s \cdot \mathit{\boldsymbol{a}}_l^{\rm{b}} \times \mathit{\boldsymbol{a}}_{\rm{s}}^{\rm{b}}$ （1）

 $\mathit{\boldsymbol{a}}_p^{\rm{b}} = s \cdot [\mathit{\boldsymbol{a}}_l^{\rm{b}} \times ]\mathit{\boldsymbol{A}}_{\rm{s}}^{\rm{b}}{\mathit{\boldsymbol{x}}_{\rm{s}}} = s \cdot \mathit{\boldsymbol{F}}{\mathit{\boldsymbol{x}}_{\rm{s}}}$ （2）

 $\mathit{\boldsymbol{a}}_p^{\rm{b}} = \left[ {\begin{array}{*{20}{l}} {\mathit{\boldsymbol{a}}_p^{\rm{b}}(1)}\\ {\mathit{\boldsymbol{a}}_p^{\rm{b}}(2)}\\ {\mathit{\boldsymbol{a}}_p^{\rm{b}}(3)} \end{array}} \right]$ （3）

 $\frac{{\mathit{\boldsymbol{a}}_p^{\rm{b}}(1){\rm{sin}}{\kern 1pt} {\kern 1pt} {\kern 1pt} \alpha - a_p^{\rm{b}}(2){\rm{cos}}{\kern 1pt} {\kern 1pt} {\kern 1pt} \alpha }}{{\mathit{\boldsymbol{a}}_p^{\rm{b}}(3)}} = \frac{{{\rm{tan}}{\kern 1pt} {\kern 1pt} {\kern 1pt} \varphi }}{{{\rm{sin}}{\kern 1pt} {\kern 1pt} {\kern 1pt} \gamma }}$ （4）

a=Fxs，则

 $\mathit{\boldsymbol{a}}_p^{\rm{b}} = s \cdot \mathit{\boldsymbol{F}}{\mathit{\boldsymbol{x}}_{\rm{s}}} = s \cdot \mathit{\boldsymbol{a}}$ （5）

 $\mathit{\boldsymbol{a}}(1){\rm{sin}}{\kern 1pt} {\kern 1pt} {\kern 1pt} \alpha {\rm{sin}}{\kern 1pt} {\kern 1pt} {\kern 1pt} \gamma - \mathit{\boldsymbol{a}}(2){\rm{cos}}{\kern 1pt} {\kern 1pt} {\kern 1pt} \alpha {\rm{sin}}{\kern 1pt} {\kern 1pt} {\kern 1pt} \gamma = \mathit{\boldsymbol{a}}(3){\rm{tan}}{\kern 1pt} {\kern 1pt} {\kern 1pt} \varphi$ （6）

 $\begin{array}{l} \left[ {\begin{array}{*{20}{l}} {\mathit{\boldsymbol{a}}(1)}&{\mathit{\boldsymbol{a}}(2)}&{\mathit{\boldsymbol{a}}(3)} \end{array}} \right]\left[ {\begin{array}{*{20}{c}} {{\rm{sin}}{\kern 1pt} {\kern 1pt} \alpha {\rm{sin}}{\kern 1pt} {\kern 1pt} \gamma }\\ { - {\rm{cos}}{\kern 1pt} {\kern 1pt} {\kern 1pt} \alpha {\rm{sin}}{\kern 1pt} {\kern 1pt} {\kern 1pt} \gamma }\\ { - {\rm{tan}}{\kern 1pt} {\kern 1pt} {\kern 1pt} \varphi } \end{array}} \right] = \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} [(\mathit{\boldsymbol{F}}{\mathit{\boldsymbol{x}}_{\rm{s}}})(1)\quad (\mathit{\boldsymbol{F}}{\mathit{\boldsymbol{x}}_{\rm{s}}})(2)\quad (\mathit{\boldsymbol{F}}{\mathit{\boldsymbol{x}}_{\rm{s}}})(3)]\left[ {\begin{array}{*{20}{c}} {{\rm{sin}}{\kern 1pt} {\kern 1pt} {\kern 1pt} \alpha {\rm{sin}}{\kern 1pt} {\kern 1pt} {\kern 1pt} \gamma }\\ { - {\rm{cos}}{\kern 1pt} {\kern 1pt} {\kern 1pt} \alpha {\rm{sin}}{\kern 1pt} {\kern 1pt} {\kern 1pt} \gamma }\\ { - {\rm{tan}}{\kern 1pt} {\kern 1pt} {\kern 1pt} \varphi } \end{array}} \right] = 0 \end{array}$ （7）

 $\mathit{\boldsymbol{d}} = \left[ {\begin{array}{*{20}{l}} {\mathit{\boldsymbol{d}}(1)}\\ {\mathit{\boldsymbol{d}}(2)}\\ {\mathit{\boldsymbol{d}}(3)} \end{array}} \right]\left[ {\begin{array}{*{20}{c}} {{\rm{sin}}{\kern 1pt} {\kern 1pt} {\kern 1pt} \alpha {\rm{sin}}{\kern 1pt} {\kern 1pt} {\kern 1pt} \gamma }\\ { - {\rm{cos}}{\kern 1pt} {\kern 1pt} {\kern 1pt} \alpha {\rm{sin}}{\kern 1pt} {\kern 1pt} {\kern 1pt} \gamma }\\ { - {\rm{tan}}{\kern 1pt} {\kern 1pt} {\kern 1pt} \varphi } \end{array}} \right]$ （8）

 ${\mathit{\boldsymbol{h}}^{\rm{T}}}\mathit{\boldsymbol{x}} = \mathit{\boldsymbol{b}}$ （9）

 ${\mathit{\boldsymbol{h}} = [\mathit{\boldsymbol{F}}(1)\mathit{\boldsymbol{d}}(1)\quad \mathit{\boldsymbol{F}}(2)\mathit{\boldsymbol{d}}(2)]}$ （10）
 ${\mathit{\boldsymbol{x}} = {{[{\rm{cos}}(\psi + {\alpha _{\rm{s}}})\quad {\rm{sin}}(\psi + {\alpha _{\rm{s}}})]}^{\rm{T}}}}$ （11）
 ${\mathit{\boldsymbol{b}} = - \mathit{\boldsymbol{F}}(3)\mathit{\boldsymbol{d}}(3)}$ （12）

 ${\left[ {\begin{array}{*{20}{l}} {{\mathit{\boldsymbol{h}}_1}}&{{\mathit{\boldsymbol{h}}_2}}& \cdots &{{\mathit{\boldsymbol{h}}_k}} \end{array}} \right]^{\rm{T}}}{\kern 1pt} {\kern 1pt} {\kern 1pt} \mathit{\boldsymbol{x}} = {\left[ {\begin{array}{*{20}{l}} {{\mathit{\boldsymbol{b}}_1}}&{{\mathit{\boldsymbol{b}}_2}}& \cdots &{{\mathit{\boldsymbol{b}}_k}} \end{array}} \right]^{\rm{T}}}$ （13）

 ${\mathit{\boldsymbol{H}}^{\rm{T}}}\mathit{\boldsymbol{x}} = {\mathit{\boldsymbol{b}}^{\rm{T}}}$ （14）

 ${(\mathit{\boldsymbol{H}} + {\mathit{\boldsymbol{E}}_{\rm{s}}})^{\rm{T}}}\mathit{\boldsymbol{x}} = {(\mathit{\boldsymbol{b}} + {\mathit{\boldsymbol{e}}_{\rm{s}}})^{\rm{T}}}$ （15）

 $(\left[ {\begin{array}{*{20}{l}} { - \mathit{\boldsymbol{b}}}&\mathit{\boldsymbol{H}} \end{array}} \right] + \left[ {\begin{array}{*{20}{l}} { - {\mathit{\boldsymbol{e}}_{\rm{s}}}}&{{\mathit{\boldsymbol{E}}_{\rm{s}}}} \end{array}} \right])\left[ {\begin{array}{*{20}{l}} 1\\ \mathit{\boldsymbol{x}} \end{array}} \right] = {\bf{0}}$ （16）

B=[-b    H], C=[-es    Es], z=[1   x]T，则式(16)可以写为

 $(\mathit{\boldsymbol{B}} + \mathit{\boldsymbol{C}})\mathit{\boldsymbol{z}} = {\bf{0}}$ （17）

 ${\rm{min}}\left\| {\mathit{\boldsymbol{Bz}}} \right\| = {\rm{min}}({\mathit{\boldsymbol{z}}^{\rm{T}}}{\mathit{\boldsymbol{B}}^{\rm{T}}}\mathit{\boldsymbol{Bz}}),{\rm{s}}{\rm{. t}}{\rm{. }}{\mathit{\boldsymbol{z}}^{\rm{T}}}\mathit{\boldsymbol{z}} = 2$ （18）

 $\psi = \left\{ {\begin{array}{*{20}{l}} {{\rm{arctan}}\left( {\frac{{{\mathit{\boldsymbol{x}}_{{\rm{ best }}}}(2)}}{{{\mathit{\boldsymbol{x}}_{{\rm{ best }}}}(1)}}} \right) - {\alpha _{\rm{s}}}}&{\psi \in [0,2\pi ]}\\ {{\rm{arctan}}\left( {\frac{{{\mathit{\boldsymbol{x}}_{{\rm{ best }}}}(2)}}{{{\mathit{\boldsymbol{x}}_{{\rm{ best }}}}(1)}}} \right) - {\alpha _{\rm{s}}} + \pi }&{\psi \in [0,2\pi ]} \end{array}} \right.$ （19）
2 多云天气条件下的大气偏振光定向方法

1) 获取任意一个像素点的偏振光定向模型随机选取一个像素点并算出该点处的高度角γo和方位角αo

 ${\rm{tan}}{\kern 1pt} {\kern 1pt} {\kern 1pt} {\gamma _{\rm{o}}} = \frac{{\sqrt {{{({x_p} - {x_{\rm{c}}})}^2} + {{({y_p} - {y_{\rm{c}}})}^2}} }}{{{f_{\rm{c}}}}}$ （20）
 ${\rm{tan}}{\kern 1pt} {\kern 1pt} {\kern 1pt} {\alpha _{\rm{o}}} = \frac{{{y_p} - {y_{\rm{c}}}}}{{{x_p} - {x_{\rm{c}}}}}$ （21）

 $\mathit{\boldsymbol{a}}_p^{\rm{b}} = s \cdot \mathit{\boldsymbol{a}}_l^{\rm{b}} \times \mathit{\boldsymbol{a}}_{\rm{s}}^{\rm{b}}$ （22）

2) 利用随机抽样一致方法筛选出符合要求的内点

3) 利用选取出的最优模型解算出航向角

3 实验验证

 图 3 偏振相机实物图 Fig. 3 Polarization camera picture

 图 4 实验装置图 Fig. 4 Experimental device picture

 图 5 少云情况 Fig. 5 Situation of partly cloudy
 图 6 多云情况 Fig. 6 Situation of cloudy

 图 7 少云情况下2种方法的误差曲线 Fig. 7 Error curves of two methods in partly cloudy situation
 图 8 多云情况下2种方法的误差曲线 Fig. 8 Error curves of two methods in cloudy situation

 方法 天气 平均误差/(°) 标准差/(°) 均方根误差/(°) 本文提出的方法 少云 -0.168 0 0.321 2 0.362 5 多云 -0.489 7 0.609 6 0.781 9 文献[20]中的方法 少云 -0.308 2 0.372 1 0.483 2 多云 -0.172 3 1.349 2 1.360 1
4 结论

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http://dx.doi.org/10.7527/S1000-6893.2020.24263

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#### 文章信息

FAN Ying, HE Xiaofeng, FAN Chen, HU Xiaoping, WU Xuesong, HAN Guoliang, LUO Kaixin

Atmospheric polarized light orientation method in cloudy weather

Acta Aeronautica et Astronautica Sinica, 2020, 41(9): 324263.
http://dx.doi.org/10.7527/S1000-6893.2020.24263