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在复眼系统中存在较多的子眼系统,视场重叠问题相对复杂,视场重叠与子眼系统的视场角和光轴夹角、观测距离相关。在不同的观测距离下,子眼系统的光轴夹角与视场角之间的关系将会导致视场重叠出现以下情况:(1)当光轴夹角不小于系统视场角之和时,复眼系统将会在观测区域内出现由盲区到重叠区域再到盲区的现象;(2)当光轴夹角小于系统光轴夹角之和时,复眼系统将出现由盲区到过重叠区域及工作区域现象。
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当光轴夹角大于视场角之和时,系统视场重叠情况如图1所示。B、Q两点将整个观测区域分为盲区和重叠区域。其中,AB、AQ分别为B、Q点在系统1光轴上的投影点距系统1中心的距离,AB、AQ表达式如下:
图 1
$\theta > ({\omega _1} + {\omega _2})$ 时系统示意图Figure 1. System schematic diagram of
$\theta > ({\omega _1} + {\omega _2})$ $$AB = \left(R - \dfrac{{R\sin {\omega _2}}}{{\sin (\theta + {\omega _2})}}\right)\left(\dfrac{{\sin ({\omega _2} + \theta )}}{{\sin ({\omega _1} + {\omega _2} + \theta )}}\right)$$ (1) $$AQ = \left(R + \frac{{R\sin {\omega _2}}}{{\sin (\theta - {\omega _2})}}\right)\left(\frac{{\sin (\theta - {\omega _2})}}{{\sin (\theta - {\omega _2} - {\omega _1})}}\right)$$ (2) 式中:
$\theta $ 为子眼间光轴夹角;R为球型固定体半径;ω1为子眼系统1的半视场角;ω2为子眼系统2的半视场角。图1表明,当光轴夹角大于系统视场角之和时,复眼系统的观测距离小于AB或大于AQ的时系统存在视场盲区,仅当观测区域处于BQ的区间内存在视场重叠,表明此种情况下系统的重叠区域仅在有限的区域内,不能满足复眼系统大视距的工作要求。
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当光轴夹角小于系统视场角之和时,系统场重叠情况如图2所示,B、E两点将整个观测区域分为盲区、过重叠区域(该区域内视场重叠率在短距离内由零增加到1)、工作区域三个区域,盲区的划分同光轴夹角大于视场角之和时的情况相同,AE为节点E在系统1光轴上的投影点距系统1中心的距离,如公式(3)所示:
图 2
$\theta \leqslant ({\omega _1} + {\omega _2})$ 时系统示意图Figure 2. System schematic diagram of
$\theta \leqslant ({\omega _1} + {\omega _2})$ $$AE = \dfrac{{2R\sin \left(\dfrac{\theta }{2}\right)\cos \left({\omega _2} - \dfrac{\theta }{2}\right)}}{{\sin (\theta + {\omega _1} - {\omega _2})}}$$ (3) 由公式(3)分析可知,当子眼见光轴夹角、子眼系统视场角固定时,过重叠区域随球型固定本体半径R的增大而增大,且与R为同一数量级。
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当子眼间的光轴夹角等于子眼系统的视场角之和时,由图2中的几何关系可知子眼系统的视场边缘线Ⅰ、Ⅱ为平行关系,即子眼间线视场的重叠区域为定量,随着观测距离的变化,子眼线视场增大,即子眼间的重叠率趋向于零。当子眼间光轴夹角小于视场角之和时,视场边缘线Ⅰ、Ⅱ将会存在一定的角度,从而保证在长工作距离下子眼间的视场重叠率趋向不为零的定值。
上述分析表明,为保证系统具有较短的过重叠区域,从而使系统在较长工作距离内具有重叠区域且不趋向于零,子眼间的光轴夹角应当小于视场角之和。
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为保证复眼系统在长工作距离下无盲区,经1.1节的分析可知,此时应当满足光轴夹角小于系统视场角之和。依据此种情况建立视场重叠计算模型,此时建立以系统1的中心为作标原点的坐标系XAY,以系统2的中心为坐标原点的坐标系X′A′Y,其中Y、Y′分别为系统1、系统2的光轴,X、X′分别为系统1,2所在位置处球型固定体的切线,其中O点为球型固定体的球心,如图3所示。当观测点M处于XAY坐标系中的坐标为(0, L)时,即观测距离为L时,子眼系统1的视场为F,H点的X坐标所形成的区域,F,H点在XAY坐标系中的X坐标分别为:
$${F_X} = L\tan {\omega _1}$$ (4) $${H_X} = - L\tan {\omega _1}$$ (5) 此时子眼系统2在观测点M所形成的系统视场为G, N点的X坐标所形成的区域,通过构建辅助点G′ N′,求出
$G,N$ 点在X′A′Y′的坐标,利用坐标X′A′Y′与坐标XAY的转换关系求得$G,N$ 点的X坐标。首先
$G'$ $N'$ 在$X'A'Y'$ 的坐标如下:$$\left\{ \begin{gathered} G{'_{X'}} = \left(\frac{{L - R}}{{\cos \theta }} + R\right)\tan {\omega _2} \hfill \\ G{'_{Y'}} = \frac{{L - R}}{{\cos \theta }} + R \hfill \\ \end{gathered} \right.$$ (6) $$ \left\{ \begin{gathered} N{'_{X'}} = - \left(\frac{{L - R}}{{\cos \theta }} + R\right)\tan {\omega _2} \hfill \\ N{'_{Y'}} = \frac{{L - R}}{{\cos \theta }} + R \hfill \\ \end{gathered} \right.$$ (7) 通过
$G'$ $N'$ 与$G,N$ 之间的关系求得X′A′Y′在X′A′Y′坐标系中的坐标如下:$$\left\{ \begin{gathered} {{G_{X'}} = \dfrac{{G{'_{X'}}\sin {\omega _2}}}{{{\rm{cos}}({\omega _2} - \theta )}}} \\ {{N_{X'}} = \dfrac{{N{'_{X'}}\sin {\omega _2}}}{{\cos (\theta + {\omega _2})}}} \end{gathered} \right.$$ (8) $$\left\{ \begin{gathered} {{G_{Y'}} = G{'_{Y'}} - G{'_{X'}}\tan \theta } \\ {N{'_{Y'}} = {N'}_{Y'} + N{'_{X'}}\tan \theta } \end{gathered} \right.$$ (9) 其中坐标
$X'A'Y'$ 与坐标$XAY$ 的转换关系如下:$$\left\{ \begin{gathered} X'\cos \theta - Y'\sin \theta + R\sin \theta = X \hfill \\ X'\sin \theta + Y'\cos \theta - R\cos \theta + R = Y \hfill \\ \end{gathered} \right.$$ (10) 利用公式(6),(7),(9),(10)求出
$G,N$ 两点在$XAY$ 坐标系中的X坐标分别为:$$\left\{ {\begin{array}{*{20}{c}} {{N_X} = \dfrac{{(R - L - R\cos \theta )\sin {\omega _2}\tan{\omega _2}}}{{\cos (\theta + {\omega _2})}} - \left[ {\dfrac{{L - R + R\cos \theta }}{{\cos \theta }} + \dfrac{{(R - L - R\cos \theta )\sin {\omega _2}\tan{\omega _2}\tan \theta }}{{\cos ({\omega _2} + \theta )\cos \theta }}} \right]\sin \theta + R\sin \theta } \\ {{G_X} = \dfrac{{(L - R + R\cos \theta )\sin {\omega _2}\tan{\omega _2}}}{{\cos \left( {{\omega _2} - \theta } \right)}} - \left[ {\dfrac{{L - R + R\cos \theta }}{{\cos \theta }} - \dfrac{{(L - R + R\cos \theta )\sin {\omega _2}\tan{\omega _2}\tan \theta }}{{\cos \theta \cos \left( {{\omega _2} - \theta } \right)}}} \right]\sin \theta + R\sin \theta } \end{array}} \right.$$ (11) 同时为保证各个系统之间不存在盲区且同时避免出现过重叠区域,由图3可知,子眼系统的视场节点应当满足:
$${F_X} > {G_X} > {H_{_X}} > {N_X}$$ (12) 若给定两个光学系统的视场角之后,为保证两个系统之间不存在盲区,同时避免过重叠现象的出现,利用公式(5),(6),(11)求得光轴夹角应当满足以下条件:
$$\left| {{\omega _{\rm{2}}}-{\omega _{\rm{1}}}} \right| < \theta < ({\omega _1} + {\omega _2})$$ (13) 在满足系统之间不存在盲区的条件下,则视场重叠定量计算如公式(14)所示:
$$\varphi (L,\theta ,R,{\omega _1},{\omega _2}) = \frac{{\left| {{H_X} - {G_X}} \right|}}{{2L\tan {\omega _1}}}$$ (14) 由于复眼系统的工作距离远远大于球型固定体的半径,根据公式(11)给出了G点X坐标随球型固定体半径的变化图像,如图(4)所示,可以看出当球型固定体半径由100 mm变化到1000 mm时G点的X坐标近似为定值,球型固定体的半径对G点坐标的影响可以忽略不计,因此简化的G点的X坐标如公式(15)所示:
$${G_X} = \frac{{L\sin {\omega _2} \tan{\omega _2}}}{{\cos \left( {{\omega _2} - \theta } \right)}} - \frac{{L\sin \theta }}{{\cos \theta }} + \frac{{L\sin {\omega _2} \tan {\omega _2}\tan \theta \sin \theta }}{{\cos \theta \cos \left( {{\omega _2} - \theta } \right)}}$$ (15) 进一步得到简化后视场重叠率定量计算公式如公式(16)所示:
$$ \varphi (L,{\omega _2},{\omega _1},\theta ) = \frac{{\left| {1 - L(\sin {\omega _2}\tan{\omega _2}\cos \theta - \sin\theta \cos ({\omega _2} - \theta ) + \sin {\omega _2}\tan{\omega _2}\tan \theta \sin \theta )} \right|}}{{2L\tan {\omega _1}\cos ({\omega _2} - \theta )\cos\theta }} $$ (16) 由公式(16)得出不同参数下复眼系统在工作区域内视场重叠率随观测距离变化图像如图(5)所示。图5(a)表明在观测距离由500 mm增大至3500 mm时,不同参数下的系统视场重叠率急剧减小,图5(b)表明当观测距离由10000 mm增大至18000 mm时,视场重叠率趋于定值。
综上所述,复眼系统设计时为避免出现视场盲区,两子眼之间的光轴夹角应当小于两子眼之间的视场角之和;此时基于子眼光轴夹角、子眼系统视场角、观测距离之间的关系,构建了子眼间视场重叠量数学模型,利用公式(16)可实现子眼间视场重叠量的精确计算,通过对该数学模型分析,为避免子眼系统间出现过重叠区域及盲区子眼间的光轴夹角应当满足公式(13);复眼系统在视场重叠区域内视场重叠量将会随着观测距离的变化逐渐出现定值,且不为零。
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文中设计了一款总视场79.23°、子眼阵列数为2个阵列、子眼数目为19个、中心子眼系统对5 km外的目标进行精准识别;边缘子眼系统完成3 km外目标的捕捉功能。综合考虑确定子眼系统光学参数如表1所示,并对子眼系统镜头进行了设计。该系统中镜头与传感器一一对应,通过球面将各个子眼进行固定,其中子眼系统的物方视场指向球心,探测系统位于球型固定体的外凸部,在满足系统大视场的条件下,缩小系统的体积,为探测系统的装调留足空间。
表 1 子眼光学参数
Table 1. Optical parameters of the subsystem
Center system Edge system ${\rm{Focal}}$/mm 54.8 19.12 ${\omega _{{x} } }{\rm{ \times } }{\omega _{{y} } }$ 4.4°×3.7° 12.5°×10.5° 建立XYZ坐标系,以子眼A、B1、C1的中心连线为X轴,子眼A、C10的中心连线为Y轴,Z轴为中心子眼A的光轴,如图6所示。在满足系统视场的前提下,尽可能的减少子眼数目、缩小系统体积,选择在观测距离3 km处,1级阵列子眼与中心子眼系统无盲区;1级阵列子眼子眼间视场重叠率在X方向和Y方向分别为75%,40%;1级阵列子眼与2级阵列子眼间视场重叠率在X方向和Y方向分别为40%,20%;1级阵列子眼与2级阵列子眼只在X方向存在视场重叠的重叠率为15%;2级阵列子眼间视场重叠率在X方向和Y方向分别为70%,30%。
利用视场重叠率的数学模型,结合公式(16),求得各个子眼间光轴夹角
$\theta $ 、${\theta _x}$ 、${\theta _y}$ 如表2所示,其中$\theta $ 、${\theta _x}$ 、${\theta _y}$ 分别为子眼间光轴夹角、子眼光轴在XZ、YZ平面中投影后与Z轴的夹角。采用圆周阵列形式进行排布,确定第一阵列子眼数目为6个,第二阵列子眼数目为12个,从而得到复眼系统整体结构。表 2 子眼间光轴夹角
Table 2. Angle of optical axis of subsystem
(A, B1) (B1, C1) (C1, C2) (B1, B2) $\theta $ 14.30° 21.60° 17.47° 14.19° ${\theta _x}$ 14.30° 21.60° 10.30° 7.04° ${\theta _y}$ 0° 0° 15.84° 12.35°
Field of view overlap rate of bionic compound eye system
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摘要: 为了实现复眼系统小型化、轻量化,提出一种基于相机外凸式的安装结构。基于该种结构,首先分析了子眼系统视场角之和与光轴夹角之间的关系,通过两子眼系统视场边缘点之间的坐标关系,建立了与子眼间光轴夹角、子眼系统视场角、观测距离相关的视场重叠率计算模型。通过对此模型分析,复眼系统中子眼间光轴夹角应当小于子眼视场角之和,且大于子眼视场角之差;在上千米的观测距离下,子眼间的视场重叠率随观测距离的变化趋向于定值。依据此模型设计了一款19孔径的复眼系统。针对3 km外目标搭建了实验,采集了子眼图像数据,实验结果表明,复眼系统实现了79.23°全视场的无盲区监测,1级阵列间子眼的实际重叠率在X、Y方向分别为71.16%、45.99%;1级与2级阵列间子眼在X、Y方向同时存在重叠时的实际重叠率分别为43.00%、18.36%,只在X方向存在重叠时的实际重叠率为14.62%;2级阵列间子眼的实际重叠率在X、Y方向分别为66.58%、24.6%。理论重叠量分别为75%、40%;40%、20%;15%;70%、30%。通过子眼实际重叠量与理论重叠量的对比分析,验证了该视场重叠计算模型的可行性。Abstract: In order to realize accurate calculation of field of view(FOV) overlap between sub eyes of compound eye system, reduce the number of sub eye systems, and make the system miniaturized and lightweight, a FOV overlap calculation model was proposed. Firstly, the relationship between the sum of FOV and the included angle of optical axis was analyzed. Based on the coordinate relationship between the edge points of FOV, a calculation model of FOV overlap rate was established, which was related to the included angle of optical axis, the angle of FOV and the observation distance. Through the analysis of this model, the angle of the optical axis between the sub eyes in the compound eye system should be less than the sum of the field angles of the sub eyes, and greater than the difference of the field angles of the sub eyes; at the observation distance of more than 3 km, the overlap rate of the field of view between the sub eyes tends to be fixed with the observation distance. According to this model, a 19 aperture compound eye system was designed. The experimental results show that the compound eye system can achieve 79.23° full field of view without blind area monitoring. The actual overlap rate of sub eyes between the first level array is 71.16% and 45.99% in the X and Y directions respectively; the actual overlap rates of the sub eyes between the first and second level arrays are 43.00% and 18.36% respectively in the X and Y directions, respectively. The actual overlap rate was 14.62% when there was overlap in direction X, and 66.58% and 24.6% in X direction and Y direction respectively. The theoretical overlaps were 75%, 40%, 40%, 20%, 15%, 70%, 30%, respectively. Through the comparative analysis of the actual overlap and theoretical overlap, the feasibility of the field overlap calculation model is verified.
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表 1 子眼光学参数
Table 1. Optical parameters of the subsystem
Center system Edge system ${\rm{Focal}}$ /mm54.8 19.12 ${\omega _{{x} } }{\rm{ \times } }{\omega _{{y} } }$ 4.4°×3.7° 12.5°×10.5° 表 2 子眼间光轴夹角
Table 2. Angle of optical axis of subsystem
(A, B1) (B1, C1) (C1, C2) (B1, B2) $\theta $ 14.30° 21.60° 17.47° 14.19° ${\theta _x}$ 14.30° 21.60° 10.30° 7.04° ${\theta _y}$ 0° 0° 15.84° 12.35° -
[1] Hu Xuelei, Gao Ming, Chen Yang. Design of curved bionic compound eye optical system with larged field of view [J]. Infrared and Laser Engineering, 2020, 49(1): 0114002. (in Chinese) doi: 10.3788/IRLA202049.0114002 [2] Zhang Jiaming, Chen Yu, Tan Haiqi. Optical system of bionic compound eye with large field of view [J]. Optics and Precision Engineering, 2020, 28(5): 1012-1020. (in Chinese) [3] Qiu Su, Ni Yu, Jin Weiqi, et al. FOV modeling of multi-aperture superposition compound eye based on micro-surface fiber faceplate [J]. Optics and Precision Engineering, 2015, 23(11): 3018-3025. (in Chinese) doi: 10.3788/OPE.20152311.3018 [4] Zhang Guang, Wang Xinhua, Li Dayu. Alignment error dertection method of sub-eye mounting hole for bionic compound eye system [J]. Chinese Optics, 2019, 12(4): 881-888. (in Chinese) [5] Zhou Tianfeng, Xie Jiaqing, Liang Zhiqiang, et al. Advances and prospesct of molding for optical microlens array [J]. Chinese Optics, 2017, 10(5): 603-618. (in Chinese) doi: 10.3788/CO.20171005.0603 [6] Brady D J, Gehm M E, Stack R A, et al. Multiscale gigapixel photography [J]. Nature, 2012, 486(7403): 386-389. doi: 10.1038/nature11150 [7] Pang Kuo, Fang Fengzhou, Song Le, et al. Bionic compound eye for 3D motion detection using an optical freeform surface [J]. Journal of the Optical Society of America B, 2017, 34(5): B28-B35. doi: 10.1364/JOSAB.34.000B28 [8] Pang Kuo, Song Le, Fang Fengzhou. Large field of view curved compound eye imaging system using optical freeform suface [J]. Journal Optoelectroncis ·Laser, 2018, 29(1): 8-13. (in Chinese) [9] Yan Fei, Guo Yunzhi, Shi Lifang. Resesrch of image mosaic algorithm based on bionic compound eye system [J]. Aero Weaponry, 2017(6): 49-53. (in Chinese) doi: 10.19297/j.cnki.41-1228/tj.2017.06.007 [10] Cao Jie, Cui Huan, Meng Lingtong, et al. Multi-resolution imaging with camera arrays on curved surface [J]. Acta Photonica Sinica, 2020, 49(4): 0411003. (in Chinese) doi: 10.3788/gzxb20204904.0411003 [11] Gao Tianyuan, Dong Zhengchao, Zhao Yu, et al. Structure and alignment of field stitching compound eye optical imaging system [J]. Acta Photonica Sinica, 2014, 43(11): 1122001. (in Chinese) doi: 10.3788/gzxb20144311.1122001 [12] Fu Yuegang, Zhao Yu, Liu Zhiying, et al. Design of the bionic compound eye optical system based on field splicing method [J]. Chinese Journal of Scientific Instrument, 2015, 36(2): 422-429. (in Chinese) [13] Fu Yuegang, Zhao Yu, Liu Zhiying, et al. Desgin of compact bionic compound eye optical system used for target identification [J]. Infrared and Laser Engineering, 2017, 46(6): 0602001. (in Chinese) doi: 10.3788/IRLA201746.0602001 [14] Tian Yuqi, Gao Tianyuan, Zhao Yu, et al. Angle error of bionic commpound eye imaging system [J]. Infrared and Laser Engineering, 2018, 47(3): 0310001. (in Chinese) doi: 10.3788/IRLA201847.0310001