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为方便分析散射介质中光传输的规律,文中首先建立散射介质中光传输的基本模型——基于Monte Carlo的Mie散射模型。Mie散射理论是由德国物理学家Gustav Mie于1908年提出的,其主要为解决光传播路径存在尺寸与光波长相当的散射颗粒时,入射光经散射颗粒散射后的散射光场分布[32]。下文以普适的单球模型为例,对Mie散射理论进行介绍。
如图1所示,设一束光强为
${I_0}$ 的自然光入射至半径为$r$ ,相对复折射率为$m = {m_r} - {\text{i}}{m_i}$ (${m_r}$ 、${m_i}$ 分别代表颗粒复折射率的实部与虚部)的各向同性散射颗粒,在距散射体$d$ 处,散射角为$\theta $ (由散射光方向矢量与入射光方向矢量间的夹角确定)的位置,散射光光强$I$ (图1中红色部分)为:$$ I = \frac{{{\lambda ^2}}}{{8{{\text{π }}^2}{d^2}}}\left[ {{{\left| {{S_1}\left( \theta \right)} \right|}^2} + {{\left| {{S_2}\left( \theta \right)} \right|}^2}} \right]{I_0} $$ (1) 公式(1)中散射振幅函数
${S_1}\left( \theta \right)$ 和${S_2}\left( \theta \right)$ 具体形式如下:$$ {S_1}\left( \theta \right) = \sum\limits_{n = 1}^\infty {\frac{{2n + 1}}{{n\left( {n + 1} \right)}}\left[ {{a_n}{\pi _n}\left( \theta \right) + {b_n}{\tau _n}\left( \theta \right)} \right]} $$ (2) $$ {S_2}\left( \theta \right) = \sum\limits_{n = 1}^\infty {\frac{{2n + 1}}{{n\left( {n + 1} \right)}}\left[ {{a_n}{\tau _n}\left( \theta \right) + {b_n}{\pi _n}\left( \theta \right)} \right]} $$ (3) 除此之外,Mie散射理论还给出了消光系数
${k_{{\text{ext}}}}$ 、散射系数${k_{{\text{sca}}}}$ 、吸收系数${k_{{\text{abs}}}}$ 等表征散射前后光场能量变化的参数:$$ {k_{{\text{ext}}}} = \frac{2}{{{\alpha ^2}}}\sum\limits_{n = 1}^\infty {\left( {2n + 1} \right){Re} \left( {{a_n} + {b_n}} \right)} $$ (4) $$ {k_{{\text{sca}}}} = \frac{2}{{{\alpha ^2}}}\sum\limits_{n = 1}^\infty {\left( {2n + 1} \right)\left( {{{\left| {{a_n}} \right|}^2} + {{\left| {{b_n}} \right|}^2}} \right)} $$ (5) $$ {k_{{\text{abs}}}} = {k_{{\text{ext}}}} - {k_{{\text{sca}}}} $$ (6) 式中:无因次直径(
$\alpha = {{{\text{π }}r} \mathord{\left/ {\vphantom {{{\text{π }}r} \lambda }} \right. } \lambda }$ )由粒径$r$ 和入射光波长$\lambda $ 确定;Mie系数中包含的${a_n}$ 和${b_n}$ 是由Ricatti-Bessel函数定义的,与散射颗粒的复折射率$m$ 有关。${\pi _n}$ 与${\tau _n}$ 则用来表征颗粒的散射强度,是关于$\cos \theta $ 的勒让德函数和一阶缔合勒让德函数。Ricatti-Bessel函数一般按无穷级数的形式给出,这给数值计算Mie散射系数带来了一定的困难。不仅如此,公式(2)、(3)中
${a_n}$ 和${b_n}$ 包含的复杂递推关系会导致Mie散射系数在计算时产生剧烈震荡[33]。考虑到Ricatti-Bessel函数会随项数增加快速收敛,因此不断有学者提出Ricatti-Bessel函数适当截断项经验公式[34-35]以及快速计算Mie散射系数的优化方法[36]。文中采用向前递推与向后递推相结合的方式,权衡计算耗时与精度,建立了一套宽范围高精度Mie散射系数的计算方法。部分计算结果如下所示:表1中,计算Mie散射系数的平均用时约为
$41.812\;{\text{ms}}$ 。文中算法相比Dave、Bohren算法的精度更高、适用范围更广,相比Lentz算法的耗时更低。验证了文中算法具备宽范围高精度的Mie散射系数计算能力,为后续散射介质传输模型的建立奠定了基础。表 1 不同方法的Mie散射系数对比表
Table 1. Comparison of Mie scattering coefficients with different methods
Particle Scattering coefficients Dave[37] Bohren[38] Lentz[34] Proposed method α=0.0001 kext 2.3068×10−17 2.3068×10−17 2.306805×10−17 2.306805×10−17 m=1.5 kabs 0 0 0 0 α=5.2182 kext 3.10542 3.10543 3.105425 3.092631 m=1.55 kabs 0 0 0 0 α=100 kext 2.0944 2.0944 2.094388 2.094388 m=1.5 kabs 0 0 0 0 α=1570.7963 kext 2.01294 — 2.012945 2.012945 m=1.342 kabs 0 — 0 0 α=25000 kext 2.00235 — 2.002352 2.002352 m=1.5 kabs 0 — 0 0 α=0.0001 kext 1.9925×10−5 1.99252×10−5 1.99252×10−5 1.99252×10−5 m=1.5-0.1 i kabs 1.9925×10−5 1.99252×10−5 1.99252×10−5 1.99252×10−5 α=5.2182 kext 2.86165 2.86165 2.861651 2.857821 m=1.55-0.1 i kabs 1.1974 1.1974 1.197404 1.197355 α=100 kext 2.0898 2.0898 2.089822 2.089822 m=1.55-0.1 i kabs 0.9577 0.9577 0.957688 0.957688 α=1570.7963 kext 2.01445 — 2.014609 2.014609 m=1.55-0.1 i kabs 0.93354 — 0.910498 0.910498 α=25000 kext 2.00232 — 2.002323 2.002323 m=1.55-0.1 i kabs 0.90641 — 0.906409 0.906409 -
对于大气环境,散射颗粒的浓度大致为0.5~50 mg/L[39]。在此类介质中传输时,绝大多数的光会被多次散射(即散射光),仅有极少的光子未被散射或只被散射了几次(即弹道光与蛇形光)。由于散射传输中存在大量随机且无序的散射事件,仅用单次散射光场分布模型已无法精确描述。基于此,文中选择Monte Carlo方法来模拟散射介质中的光传输过程。如图2所示仿真过程,首先引入光子团的概念,光子团是指:一定数量状态完全相同的光子集合,且该集合内光子传播的行为当且仅当与散射颗粒作用时才会产生差异[40]。基于图1所示的单次散射模型,计算单次反照率(光子团由散射引起的能量损失率)
${\omega _0}$ ,以及散射自由程$l$ 、散射角$\theta $ 、方位角$\;\beta $ 的概率分布[36,41-42],便可利用Monte Carlo方法来模拟散射介质中的光传输过程。由于穿透激光雷达系统旨在获取散射介质中目标的时空信息,文中对Mie散射模型做出了改进,在散射介质中置入待测目标,并精确统计光子团的飞行时间。过程中需计算光子团与目标的作用坐标,即光子飞行路径所在直线的参数方程与目标法向表面微元的交点,接下来按照光子团入射坐标及方向余弦抽样,光子脉冲抽样,光子自由程抽样,光子团散射角、方位角抽样,最后按照目标碰撞假定、光子团能量、光子团飞行路程累加的流程完成Monte Carlo模拟。
由于该仿真模型中所用抽样的精度依赖于Mie散射系数的计算能力,因而仿真模型同样具有宽范围散射介质模拟、高仿真精度的特点。借此仿真模型,可对不同种类,不同强度的散射介质;不同深度,不同材质的目标;不同范围,不同空间采样率的激光雷达进行探测模拟,并获取目标的强度图、深度图、时间直方图,对后续研究具有显著的指导作用。
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对比图3(a)、(b)得出:即使目标在强度分布的特征难以辨别,但在时间信号相应位置处仍存在突出的目标波形。此时引入目标距离信息,再借助时间选通等方式便能较好的提取目标特征。然而,这种提取手段的前提是散射介质的后向散射与目标峰形不产生混叠,或目标峰远高于临近通道的光子计数。在实际环境中,散射介质
${\text{OT}}$ 的提升往往伴随着目标深度或散射颗粒浓度的增加。若目标深度不变,所处环境的散射颗粒浓度增加,目标峰则可能与介质的后向散射信号产生混叠,如图4所示。图 4 目标峰与介质的后向散射信号产生混叠的仿真结果
Figure 4. Simulation results of aliasing between the target peak and the backscattered signal of the medium
对于散射程度较小的大尺度探测环境,常规算法通常能够有效地提取目标信息。但该类算法的恢复效果较依赖于信号中弹道光的信噪比,且通常不考虑散射介质的传输特性,难以应用于散射更强的环境。散射介质中散射颗粒浓度提高会增强介质的后向散射;抑制目标峰的幅值;目标峰最终淹没于后向散射信号难以被采集。对于存在混叠的散射场景,常规算法便难以适用。此时需从散射介质的传输特性入手,以有效提取目标信息。对此,文中提出一种基于信号时域分布的自适应目标提取算法:
首先,设采集信号中介质后向散射满足的分布为
${S_{\text{b}}}$ ,目标峰满足的分布为${S_{\text{o}}}$ ,则采集信号$S$ 满足:$$ S\left( t \right) = {c_{\text{b}}}{S_{\text{b}}}\left( t \right) + {c_{\text{o}}}{S_{\text{o}}}\left( t \right) + {s_{\text{n}}}\left( t \right) $$ (7) 式中:
${c_{\text{b}}}$ 、${c_{\text{o}}}$ 代表叠加权重;$ {s_{\text{n}}}\left( t \right) $ 表示采集信号中的随机噪声。$ {c_{\text{b}}}{S_{\text{b}}} $ 由Mie理论给出,散射介质中光子团的散射自由程应满足:$$ {f_0}\left( l \right) = \left\{ \begin{gathered} \frac{1}{{\bar l}}\exp \left( { - \frac{1}{{\bar l}}l} \right),\;l > 0 \\ 0\quad \quad \quad \quad \quad ,\;l \leqslant 0 \\ \end{gathered} \right. $$ (8) 式中:
$\bar l$ 表示当前散射环境的平均自由程。由于碰撞次数相同的光子团在接收信号中应服从相同的分布,且Monte Carlo过程的光子团互相独立。因而碰撞次数相同的光子团的叠加应服从伽马分布,即总计发生$i$ 次碰撞的接收信号满足时间分布:$$ {f_i}\left( {t,\bar t,i} \right) = \frac{{{{\bar t}^{ - i}}}}{{\varGamma \left( i \right)}}{t^{i - 1}}\exp \left( { - \frac{1}{{\bar t}}t} \right),\;t > 0 $$ (9) 式中:
$\bar t$ 为散射环境中两次散射事件的平均时间间隔,满足$\bar t = {{\bar l} \mathord{\left/ {\vphantom {{\bar l} {\text{c}}}} \right. } {{c}}}$ (${{c}}$ 为光速)。借助Monte Carlo仿真模型,可以统计出接收信号中不同碰撞次数的光子团满足的频数序列(
$N = {n_1},{n_2}, \cdots ,{n_i}, \cdots $ ,式中$i$ 表示光子团发生碰撞的次数,${n_i}$ 表示总计发生$i$ 次碰撞的光子团个数,且$i \to \infty $ )。实际过程中的单次反照率令光子团能量将随碰撞次数指数递减,若选择最大碰撞次数$a$ 作为次数统计的上限,可以最大似然的估计出发生$i$ 次碰撞的光子团在接收信号中出现的频率为:$ {p_i} = {{{n_i}} \mathord{\left/ {\vphantom {{{n_i}} {\sum\nolimits_{i = 1}^a {{n_i}} }}} \right. } {\sum\nolimits_{i = 1}^a {{n_i}} }} $ ,那么采集信号中后向散射的分布${S_{\text{b}}}$ 应为[30]:$$ {S_{\text{b}}} = \sum\limits_{i = 1}^a {{p_i}{f_i}\left( {t,\bar t,i} \right)} $$ (10) 若发射系统的输入功率为
${P_{{\text{in}}}}$ ,则${c_{\text{b}}}{S_{\text{b}}}$ 为:$$ {c_{\text{b}}}{S_{\text{b}}} = {P_{{\text{in}}}}\sum\limits_{i = 1}^a {\omega _0^i{p_i}{f_i}\left( {t,\bar t,i} \right)} $$ (11) 式中:
${\omega _0}$ 为散射颗粒的单次散射反照率。在已知
$ {c_{\text{b}}}{S_{\text{b}}} $ 的基础上,若通过滤波对信号$S$ 的噪声加以滤除,就能获得目标信号$ {c_{\text{o}}}{S_{\text{o}}} $ ,例如采取小波变换等方式处理信号中的${s_{\text{n}}}$ 部分。为考察该算法是否有效,文中首先利用仿真数据进行验证,下图为一组时间直方信号的去噪结果,包含后向散射信号${S_{\text{b}}}$ 分布统计拟合结果与差分信号的目标峰提取结果:据公式(7),对去噪信号与
${c_{\text{b}}}{S_{\text{b}}}$ 进行差分便可得到${c_{\text{o}}}{S_{\text{o}}}$ 。但由于小波变换等去噪算法不能理想的剔除系统噪声,差分后的结果一般为$ {c_{\text{o}}}{S_{\text{o}}} + \delta \left( {{s_{\text{n}}}} \right) $ ,这里$ \delta \left( {{s_{\text{n}}}} \right) $ 表示经由小波变换后残余的噪声。考虑目标信噪比趋于
$1$ 的情况:目标峰处于后向散射信号拖尾、且强度较低的部分,差分过程对目标峰型引入的变形较小。因$\delta \left( {{s_{\text{n}}}} \right)$ 与${S_{\text{o}}}$ 存在分布差异,可采用相关的方法提取${c_{\text{o}}}{S_{\text{o}}}$ ,并得出目标峰位及标准差,完成目标的提取(图5 (b))。图 5 时间直方信号的去噪结果。(a) 后向散射信号
${S_{\text{b}}}$ 分布统计拟合结果;(b) 差分信号的目标峰提取结果Figure 5. Denoising results of time histogram signal. (a) Distribution fitting result of backscattering signal
${S_{\text{b}}}$ ; (b) Target peak extraction result for differential signal综上,该算法能够自适应地提取出目标特征,算法平均耗时为
$1.023{\text{s}}$ ,对目标强度图像的信背比(目标强度与背景平均强度的比)与边缘锐度带来较大提升。为进一步验证该算法的适用范围,文中对两种介质的仿真信号进行了特征提取,结果如下:图6、图7所示为散射介质中激光雷达系统的仿真探测信号的强度图与时间直方图。两组图的仿真参数如下:图6中,目标距离接收孔径
$0.2\;{\text{m}}$ 、散射颗粒不对称因子$0.924$ 、从图(a)~(c)可知目标处${\text{OT}}$ 分别为:$4.993\;6$ 、$6.048\;5$ 、$6.912\;3$ ,算法平均耗时为$1.085\;{\text{s}}$ ;图7中,目标距离接收孔径1 m、散射颗粒不对称因子$0.524$ 、从图(a)~(c)可知目标处${\text{OT}}$ 分别为:$1.920\;3$ 、$3.072\;6$ 、$3.936\;4$ ,算法平均耗时为$1.117\;{\text{s}}$ 。对比两图结果,尽管在${\text{OT}}$ 较大时该算法恢复目标的强度较低,但仍较好地保留了目标特征。相比于恢复前的强度分布,目标信背比与目标特征的留存度都有了较大的提升。通过仿真数据的验证,证明了该算法的去散射能力与信号峰的信噪比直接相关。该算法的实用性依靠实验进一步验证。
Research on penetrating imaging LIDAR based on time-correlated single photon counting (invited)
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摘要: 激光雷达是一种能够快速获取目标三维信息的光学传感器,凭借精确定位与高效识别能力,其在军事侦察、无人驾驶、空间交会对接等领域发挥着越来越重要的作用。然而在浓雾、烟尘、海水等复杂的气候或环境下,传输介质对光场的强烈散射作用会造成传统激光雷达接收信号的严重退化,致使其性能迅速降低,甚至根本无法正常使用。针对该类环境下接收信号的退化特性,首先建立了Monte Carlo的Mie散射瞬态光场传输模型,接着通过计算机模拟仿真了传输光场的时域分布规律,针对该规律研究了时域信号去散射效应的滤波算法,最后在实验室搭建了基于时间相关单光子计数的穿透成像激光雷达系统,开展了雾气模拟环境下的成像实验,结果验证了该穿透成像激光雷达系统的处理方法对目标图像重构质量的提高具有较好的效果。该研究为激光雷达在复杂散射环境中的进一步应用提供了基础。Abstract: Light detection and ranging (LIDAR) is a kind of optical sensor with accurate positioning and efficient identification ability that can quickly acquire three-dimensional information of targets. Therefore, LIDAR plays an increasingly important role in military reconnaissance, unmanned driving, space docking and other fields. However, in complex environment such as fog, smoke, sea and so on, scattering effect in light field causes serious degradation of the received signal in traditional LIDAR. Under these environmental conditions, the performance of traditional LIDAR will decrease rapidly, or fail to work. Aiming at the degradation characteristics of received signal in scattering environment, the transmission model of Mie scattering transient light field of Monte Carlo was firstly established. Then, the time-domain distribution law of the transmission light field was simulated by computer software. According to this law, the filtering algorithm of the time-domain signal de-scattering effect was studied. Finally, a kind of penetrating LIDAR based on TCSPC was built in laboratory. Through the imaging experiments in fog simulation environment, the results verify that the penetrating LIDAR method has a good effect on improving the quality of target image reconstruction. This study provides a base for the further applications of LIDAR in complex scattering environment.
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Key words:
- LIDAR /
- penetrating imaging /
- Mie scattering /
- TCSPC
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图 12 多目标实验结果。(a) 场景示意图;(b) 采集信号及算法拟合结果;(c) 算法处理后的强度图(右下为原始强度图);(d) 算法处理后的深度图(单位:m)
Figure 12. Results of multi-objective experiments. (a) Scene graph; (b) Received signals and fitting results; (c) The intensity image after algorithm processing (bottom right is the raw intensity image); (d) The depth image after algorithm processing (unit: m)
表 1 不同方法的Mie散射系数对比表
Table 1. Comparison of Mie scattering coefficients with different methods
Particle Scattering coefficients Dave[37] Bohren[38] Lentz[34] Proposed method α=0.0001 kext 2.3068×10−17 2.3068×10−17 2.306805×10−17 2.306805×10−17 m=1.5 kabs 0 0 0 0 α=5.2182 kext 3.10542 3.10543 3.105425 3.092631 m=1.55 kabs 0 0 0 0 α=100 kext 2.0944 2.0944 2.094388 2.094388 m=1.5 kabs 0 0 0 0 α=1570.7963 kext 2.01294 — 2.012945 2.012945 m=1.342 kabs 0 — 0 0 α=25000 kext 2.00235 — 2.002352 2.002352 m=1.5 kabs 0 — 0 0 α=0.0001 kext 1.9925×10−5 1.99252×10−5 1.99252×10−5 1.99252×10−5 m=1.5-0.1 i kabs 1.9925×10−5 1.99252×10−5 1.99252×10−5 1.99252×10−5 α=5.2182 kext 2.86165 2.86165 2.861651 2.857821 m=1.55-0.1 i kabs 1.1974 1.1974 1.197404 1.197355 α=100 kext 2.0898 2.0898 2.089822 2.089822 m=1.55-0.1 i kabs 0.9577 0.9577 0.957688 0.957688 α=1570.7963 kext 2.01445 — 2.014609 2.014609 m=1.55-0.1 i kabs 0.93354 — 0.910498 0.910498 α=25000 kext 2.00232 — 2.002323 2.002323 m=1.55-0.1 i kabs 0.90641 — 0.906409 0.906409 -
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