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衍射光谱成像系统使用的衍射光学元件是多台阶衬底结构的高衍射效率元件,有利于光学成像系统实现微型化、轻小型化和高集成化。双通道衍射计算光谱成像系统包括衍射成像光谱系统以及全色灰度相机成像系统。衍射光谱成像系统的核心主要部件是衍射成像透镜,在对入射光线聚焦成像的同时还能产生轴向的色差。衍射光学具有独特的色散特性,色散与波长有关[9-13]。衍射光学元件不同波长焦距与入射光波长成反比:
$$ f(\lambda ) = \frac{{{\lambda _0}{f_0}}}{\lambda } $$ (1) 式中:f0为设计波长λ0的焦距,即设计焦距。
如图1所示,衍射成像光谱系统的色散方向是沿着光纤传播方向,在色散空间上产生一定数量的光谱切片,因此光谱切片的序列方向即为光轴方向。在光轴方向上,不同切片处的点扩展函数PSF表示相应准焦和离焦光谱切片的成像效应、清晰和模糊的程度,PSF对光谱分辨率、带宽起到决定性作用。
光谱分辨率是光谱仪系统的重要参数,用f/#表示系统的F数,则波长λ的光谱分辨率表达式为:
$$ \Delta \lambda = \frac{{{\lambda ^2}}}{{{\lambda _0}{f_0}}}\Delta = \frac{{{\lambda ^2}}}{{{\lambda _0}{f_0}}} \cdot 8\lambda {({f \mathord{\left/ {\vphantom {f \# }} \right. } \# })^2} = \frac{{8{\lambda _0}\lambda }}{{{f_{_0}}}}{({f \mathord{\left/ {\vphantom {f \# }} \right. } \# })^2} $$ (2) 衍射效率是衍射成像系统里的关键指标,针对指定的位相函数,在制备微结构单元衍射面时,衍射面的第m级次的衍射效率计算方法如公式(3),N为衍射元件DOE的级数,当N取6时衍射效率接近90%[9-10]。
$$ \eta _m^N = \frac{1}{{{N^2}}} \cdot \dfrac{{{{\sin }^2}\left[ {\pi \left( {1 - \dfrac{{{\lambda _0}}}{\lambda }} \right)} \right]}}{{{{\sin }^2}\left[ {\dfrac{\pi }{N}\left( {1 - \dfrac{{{\lambda _0}}}{\lambda }} \right)} \right]}} \cdot {{{\rm{sinc}}} ^2}\left( {\dfrac{m}{N}} \right) $$ (3) -
衍射光谱系统成像的光谱重构有很多种方法[5-11],文中针对光谱图像函数建立量化模型,将光谱图像数据立方体看作x,y,z(λ)的连续函数,图像在x,y方向经过面阵探测器采样,探测器沿光轴z方向在不同的光谱聚焦点位置进行扫描成像。数字量化表征目标信号图像函数,用h()表示点扩散函数。三维矩阵方程如下:
$$ \begin{gathered} i\left( {{x_i},{y_i},{z_i}} \right) = \hfill \\ \int\limits_{ - \infty }^\infty {\int\limits_{ - \infty }^\infty {\int\limits_{ - \infty }^\infty {o\left( {x,y,z} \right)} } } h\left( {{x_i} - x,{y_i} - y,{z_i} - z} \right){\rm{d}}x{\rm{d}}y{\rm{d}}z \hfill \\ \end{gathered} $$ (4) 对x、y、z进行离散化处理,可将方程(4)的积分形式改写为求和形式,以四种光谱色散为例,用o(x, y)表示每个光谱位置的准焦图像,h(x, y)表示相应的准焦和离焦点扩散函数,探测器分别在四种光谱位置探测,得到的四种信号能量可表示为:
$$ \begin{split}&{i}_{\text{1}}\left({x}\text{,}{y}\right)\text={h}_{11}\left({x}\text{,}{y}\right)\ast {o}_{1}\left({x}\text{,}{y}\right)+{h}_{12}\left({x}\text{,}{y}\right)\ast {o}_{2}\left({x}\text{,}{y}\right) +\\&{h}_{13}\left({x}\text{,}{y}\right)\ast {o}_{3}\left({x}\text{,}{y}\right)+{h}_{14}\left({x}\text{,}{y}\right)\ast {o}_{4}\left({x}\text{,}{y}\right)\\& {i}_{2}\left({x}\text{,}{y}\right)\text={h}_{21}\left({x}\text{,}{y}\right)\ast {o}_{1}\left({x}\text{,}{y}\right)+{h}_{22}\left({x}\text{,}{y}\right)\ast {o}_{2}\left({x}\text{,}{y}\right) +\\&{h}_{23}\left({x}\text{,}{y}\right)\ast {o}_{3}\left({x}\text{,}{y}\right)+{h}_{24}\left({x}\text{,}{y}\right)\ast {o}_{4}\left({x}\text{,}{y}\right)\\& {i}_{3}\left({x}\text{,}{y}\right)\text={h}_{31}\left({x}\text{,}{y}\right)\ast {o}_{1}\left({x}\text{,}{y}\right)+{h}_{32}\left({x}\text{,}{y}\right)\ast {o}_{2}\left({x}\text{,}{y}\right) +\\&{h}_{33}\left({x}\text{,}{y}\right)\ast {o}_{3}\left({x}\text{,}{y}\right)+{h}_{34}\left({x}\text{,}{y}\right)\ast {o}_{4}\left({x}\text{,}{y}\right)\\& {i}_{4}\left({x}\text{,}{y}\right)\text={h}_{41}\left({x}\text{,}{y}\right)\ast {o}_{1}\left({x}\text{,}{y}\right)+{h}_{42}\left({x}\text{,}{y}\right)\ast {o}_{2}\left({x}\text{,}{y}\right) +\\&{h}_{43}\left({x}\text{,}{y}\right)\ast {o}_{3}\left({x}\text{,}{y}\right)+{h}_{44}\left({x}\text{,}{y}\right)\ast {o}_{4}\left({x}\text{,}{y}\right)\end{split} $$ (5) 上述方程描述了探测器在每个位置处探测得到的混叠光谱信号,每一光谱目标都与对应位置处的点扩展函数卷积,混叠光谱图像可以看作是所有卷积叠加的结果。
能量表达式(5)经过傅里叶变换后,用矩阵表达的形式为:
$$ \begin{split}&\left[\begin{array}{c}{I}_{\text{1}}\left(\xi \text{,}\zeta \right)\\ {I}_{\text{2}}\left(\xi \text{,}\zeta \right)\\ {I}_{\text{3}}\left(\xi \text{,}\zeta \right)\\ {I}_{\text{4}}\left(\xi \text{,}\zeta \right)\end{array}\right]\text=\\ &\left[\begin{array}{cccc}{H}_{11}\left(\xi \text{,}\zeta \right)& {H}_{12}\left(\xi \text{,}\zeta \right)& {H}_{13}\left(\xi \text{,}\zeta \right)& {H}_{14}\left(\xi \text{,}\zeta \right)\\ {H}_{21}\left(\xi \text{,}\zeta \right)& {H}_{22}\left(\xi \text{,}\zeta \right)& {H}_{23}\left(\xi \text{,}\zeta \right)& {H}_{24}\left(\xi \text{,}\zeta \right)\\ {H}_{31}\left(\xi \text{,}\zeta \right)& {H}_{32}\left(\xi \text{,}\zeta \right)& {H}_{33}\left(\xi \text{,}\zeta \right)& {H}_{34}\left(\xi \text{,}\zeta \right)\\ {H}_{41}\left(\xi \text{,}\zeta \right)& {H}_{42}\left(\xi \text{,}\zeta \right)& {H}_{43}\left(\xi \text{,}\zeta \right)& {H}_{44}\left(\xi \text{,}\zeta \right)\end{array}\right] \times \\ & \left[\begin{array}{c}{O}_{1}\left(\xi \text{,}\zeta \right)\\ {O}_{2}\left(\xi \text{,}\zeta \right)\\ {O}_{3}\left(\xi \text{,}\zeta \right)\\ {O}_{4}\left(\xi \text{,}\zeta \right)\end{array}\right]\end{split} $$ (6) 式中:I1、I2、I3、I4是探测器在不同光谱位置处探测得到的信号。不同光谱位置处的准焦和离焦点扩散函数H()可通过系统标定得到,通过逆运算即可求得光谱信号O1、O2、O3、O4。
$$ O\left( {\xi ,\zeta } \right) = {H^{ - 1}}\left( {\xi ,\zeta } \right)I\left( {\xi ,\zeta } \right)$$ (7) 实际定标过程存在误差,系统的色散参数难以完全准确测试,并且成像过程存在噪声,这些都会带来解算误差。通过增加一路全色通道,给解算过程提供附加信息,提升解算精度。全色信息为SPAN,全色信息是所有光谱信息叠加后的总信息,即SPAN= O1+O2+O3+O4,可作为附加信息进行联合解算。
每种离焦量有相应的点扩展函数,对点扩展函数进行傅里叶变换,将变换后的点扩展函数组成块矩阵。当光谱通道数量较多、离焦量较大时,矩阵中的某些块近似为零,导致矩阵奇异。求解奇异矩阵的逆矩阵是解决线性最小二乘问题,可采用SVD奇异值分解方法,任意M×N矩阵,行数M≥列数N,进行奇异值分解的表达式为:
$$ {H^{ - 1}} = V\left[ {diag(1/{w_i})} \right] \cdot {U^{\rm{T}}} $$ (8) 式中:w为奇异值。对w进行变换,附加规则滤波器,可有效减小系统噪声对解算的影响。将1/w变换为:
$$ {1 \mathord{\left/ {\vphantom {1 w}} \right. } w} = {w \mathord{\left/ {\vphantom {w {\left( {{w^2} + {a^2}} \right)}}} \right. } {\left( {{w^2} + {a^2}} \right)}}$$ (9) 将SVD代入重建方程(7),可得到:
$$ \begin{split} O(\xi ,\zeta ) =& V(\xi ,\zeta ) \times \left[ {diag\left( {\frac{{w(\xi ,\zeta )}}{{{w^2}(\xi ,\zeta ) + {\alpha ^2}}}} \right)} \right] \times \\ &{U^{\rm{T}}}(\xi ,\zeta ) \times I(\xi ,\zeta ) \end{split} $$ (10)
Double channels diffractive computational imaging spectrometer system
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摘要: 常规衍射光谱成像系统采用单通道方案,主要针对简单图形目标或光谱特征已知的气体目标进行模拟仿真和光谱成像实验。而当目标为自然场景等复杂景物时,成像系统的光谱解算效果和精度难以保证。针对复杂景物成像,设计了一套双通道可见近红外衍射计算成像光谱仪系统,在常规单通道衍射成像光谱仪系统的基础上,增加一路全色相机成像,可以为衍射成像通道提供复杂景物的全色信息和先验知识。将两个通道的数据进行联合处理,提升最终的光谱数据反演效果和反演精度。介绍了系统组成和基本原理,分析了系统性能,利用仿真程序模拟了系统成像过程。在实验室搭建了可见近红外衍射计算成像光谱原理验证装置。对实验得到的450~800 nm的可见近红外混叠光谱数据进行光谱复原。利用海洋光学光谱仪测试色板的光谱曲线作为标准谱线,与复原得到的光谱数据进行对比,反演的光谱数据平均精度优于90%。通过衍射计算成像原理分析、模拟仿真和原理实验,验证了双通道衍射计算成像系统原理的正确性,能够反演得到精度优于90%的复杂景物光谱数据,提升了衍射成像光谱系统应用潜力和应用价值。Abstract: The conventional diffraction spectrum imaging system adopts the single channel scheme, which mainly carries out simulation and spectral imaging experiments for simple graphic targets or gas targets with known spectral characteristics. When the target is a complex scene such as natural scene, the spectral solution effect and accuracy of the imaging system are difficult to ensure. For the imaging of complex scenery, a dual channel visible and near-infrared diffraction computational imaging spectrometer system was designed. Based on the conventional single channel diffraction imaging spectrometer system, adding a panchromatic camera imaging coluld provide panchromatic information and a priori knowledge of complex scenes for diffraction imaging channels. The data of the two channels were jointly processed to improve the final spectral data inversion effect and inversion accuracy. The system composition and basic principle were introduced, the system performance was analyzed, and the imaging process of the system was simulated by using the simulation program. A verification device for the principle of visible and near-infrared diffractive computational imaging spectrometer system was built in the laboratory. Spectral restoration was carried out on the visible and near-infrared aliasing spectral data of 450-800 nm. Using the spectral curve of the color plate tested by ocean optics spectrometer as the standard spectral line, compared with the restored spectral data, the average accuracy of the retrieved spectral data was better than 90%. Through theoretical analysis, system simulation and imaging experiment, the correctness and feasibility of the system principle were verified. It can obtain better spectral solution effect and accuracy of complex scenery, and improve the application potential and application value of diffraction imaging spectral system.
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Key words:
- spectral /
- diffractive /
- computational imaging /
- image restoration
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