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作为一种新型的间接成像方式,关联成像又称“鬼成像”,在过去的20年里一直是光学成像中的一个研究热点。1995年,史砚华等人利用自发参数下转换产生的量子纠缠光子对,在实验上观察到了第一个鬼成像[23-24]。2002年,Boyd等人利用经典光源在实验上实现了鬼成像,证明了鬼成像并非量子光场独有[25-26]。传统鬼成像系统需要两束空间相关联光束,一束不经过物体,称为参考光,用来记录光的空间分布;另一束为物光,和目标物体相互作用后,其光强值被一个单像素探测器记录。两束光并不能单独成像,但通过关联运算可以计算得到目标像,即称为鬼像。
随着空间光调制器(SLM)的出现,2008年Shapiro提出了计算鬼成像[27]。在计算鬼成像中,参考光被一个空间光调制器替代,即实验中只需一路被调制后的光束和一个单像素探测器即可恢复物体的像,光路明显简化,如图1(a)所示。同年更早时间,Duarte等人利用数字微镜器件(DMD)配合一个单像素探测器,结合物体信号本身的稀疏特性,通过压缩感知算法计算恢复出了物体的像,这被称为单像素成像,实验光路如图1(b)所示[2]。通过对比图1(a)和图1(b)可以看到,计算鬼成像与单像素成像的不同仅在于物体与掩模图案生成器的前后位置上。而两者的其他方面,例如光源、探测器和图像恢复算法等都可以互通。随着研究的不断深入,研究者普遍认为计算鬼成像和单像素成像本质相同,因此以下计算鬼成像和单像素成像将被视为同一个概念。
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单像素成像通过将一系列空间分布不同的掩模图案与目标物体作用,之后利用没有空间分辨率的单像素探测器探测作用后的光强值,进而可以通过不同的算法计算两者之间的关联,从而恢复出目标图像[28]。单像素探测实际上是在进行多态扫描,即用一个像素来感知同时来自多个位置的强度信息的叠加。相比于扫描成像,单像素成像的成像速度更快,更加适用于商用。
用数学的方法理解单像素成像的过程:假定目标图像是一个灰度图,其可以视为一个二维矩阵T,矩阵中的每一个元素都代表着相应图像中的像素值。同理,用空间光调制器调控入射光得到的N个掩模图像也可以视为矩阵
$ {\left\{{M}^{m\times n}\right\}}_{N} $ 。用不具有空间分辨能力的单像素相机来测量N个掩模图案$ {\left\{{M}^{m\times n}\right\}}_{N} $ 和目标图像T相互作用后的光强信息$ {\left\{Io\right\}}_{N} $ ,可表示为:$$ I{o}_{k}={\sum }_{m,n}{{T}_{}}^{m\times n}\cdot {{M}_{k}}^{m\times n} \text{} $$ (1) 式中:
$ I{o}_{k} $ 表示第k次入射的掩模图像和目标图像相互作用之后单像素探测器测得的光强值。重构图像的过程就是利用$ {\left\{Io\right\}}_{N} $ 和$ {\left\{{M}^{m\times n}\right\}}_{N} $ 求解T中包含的N个未知数的问题。将掩模$ {\left\{{M}^{m\times n}\right\}}_{N} $ 和对应的光强值$ {\left\{Io\right\}}_{N} $ 做关联运算,最简单的鬼成像(ghost imaging, GI)算法由公式(2)的二阶关联运算定义给出[29-30]:$$ {G}^{\left(2\right)}=\frac{1}{N}{\sum }_{k=1}^{N}\left(I{o}_{k}-\left\langle{{I}_{O}}\right\rangle\right)\cdot {({M}_{k}}^{m\times n}-\left\langle{{M}^{m\times n}}\right\rangle) \text{} $$ (2) 式中:G(2)代表重构的目标图像;
$ \left\langle{{I}_{O}}\right\rangle $ 和$ \left\langle{{M}^{m\times n}}\right\rangle $ 分别表示N次测量物光和信号光的平均值。进一步为了提高成像质量,差分鬼成像(differential GI, DGI)的算法被提出[31],与GI定义的不同在于其物光$ {I}_{O} $ 被以下差分形式所替代:$$ {I}_{k}^{DGI}=I{o}_{k}-\frac{\left\langle{{I}_{O}}\right\rangle}{\left\langle{{M}_{k}}\right\rangle}{M}_{k} \text{} $$ (3) 式中:
$ {M}_{k} $ 代表的是$ {{M}_{k}}^{m\times n} $ 矩阵的总和;$ \left\langle{{M}_{k}}\right\rangle $ 表示N个掩模图像的平均值。分析可知,最简单的$ {{M}_{k}}^{m\times n} $ 是只有一个元素为1、其余均为0的矩阵,与光栅扫描成像完全相同。考虑到许多自然场景都是稀疏或可压缩的,可以用正交性更好的掩模图像来获取物体信息,如哈达马(Hadamard)、傅里叶(Fourier)、小波分析(wavelet)等。考虑到成像物体的稀疏特性,子采样(sub-sampling)的策略被引入到单像素成像中,即在对包含N个像素的目标成像时只测量M次(M<N),称为压缩鬼成像(compressive ghost imaging, CGI)。在压缩感知算法中,$ {{M}_{k}}^{m\times n} $ 变形为一个行矢量($ 1\times K $ ,$ K=m\times n $ ),$ I{o}_{k} $ 变形为一个列矢量$ {I}^{CGI}(N\times 1 $ ),N次测量所使用的掩模图像$ {\left\{{M}^{m\times n}\right\}}_{k} $ 整形为一个二维矩阵A($ N\times K $ )。假定图像是稀疏的,则目标图像可以通过如下凸优化步骤进行重构[32-33]:$$ {T}^{CGI}=\left|T\right|, min {‖T‖}_{1}\;\;满足{I}^{CGI}=AT $$ (4) 式中:
$ {T}^{CGI} $ 为重构图像;T为目标图像;$ {‖T‖}_{1} $ 为求T的$ {L}_{1} $ 范数。CGI算法相较于GI和DGI的测量次数明显减少,成像效率明显提高,因而在实验中被广泛使用。更多的单像素成像重构算法可以参考参考文献[34]。
Single-pixel imaging and metasurface imaging (Invited)
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摘要: 单像素成像作为一种典型的计算成像方式,利用单像素探测器测量一系列掩模图像照射目标之后的光强值,进而通过不同重构算法恢复目标图像。相较于多像素探测器(CCD或CMOS),单像素成像克服了硬件的限制,在某些特殊波段,探测效率更高,响应更快。超表面是由亚波长的金属或介质单元构成的二维人造结构。在可见光波段,超表面可以调控入射光的多种自由度以实现多通道全息图的显示;在微波波段,超表面可以与导波模式进行耦合辐射出多种模式图案。简单介绍了单像素成像的研究背景、成像原理和重构算法、超表面成像的研究背景,并主要讨论了两种成像方式在可见光波段以及微波波段的结合以及相关研究,最后提出了未来的发展方向。Abstract: As a typical computational imaging modality, single-pixel imaging uses a single-pixel detector to measure the light intensities reflected or transmitted from the target after its interaction with a series of patterns. By calculating the correlation of the measured intensities and relevant patterns with different reconstruction algorithms, the target image can be recovered. Compared with multi-pixel detector (i.e. CCD or CMOS), single-pixel imaging overcomes hardware limitations and the detection efficiency is higher, and the response is faster in some special wavebands. Metasurfaces are a kind of artificial two-dimensional materials consisting of an array of subwavelength metallic or dielectric unit cells. In the optical wavelength regime, the metasurface can display various holograms by adjusting different degrees of freedom of incident light. In the microwave regime, the metasurface can couple with the waveguide and emit various radiating modes as patterns. The research background, imaging principle, reconstruction algorithms of single-pixel imaging, and the research background of metasurface imaging were reviewed. The discussion of relevant works was mainly focused on the combination of single-pixel imaging and metasurface imaging in optical and microwave regimes, and finally a perspective on the potential development in future was proposed.
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Key words:
- single-pixel imaging /
- metasurface /
- computational imaging
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图 3 超表面成像作为目标图像。 (a)实验装置示意图;(b)超表面单元结构构成示意图;(c)扫描电子显微镜下超表面部分区域结构图;(d)左旋和右旋偏振光入射下超表面全息成像;(e)不同鬼成像算法仿真结果的图像对比;(f)不同鬼成像算法实验结果的图像对比;(g)基于手性超表面全息成像和压缩鬼成像的光学加密[65]
Figure 3. Metasurface imaging as target image. (a) Schematic diagram of experimental setup; (b) Schematic diagram of unit cell of the metasurface; (c) Scanning electron microscopy image of the metasurface; (d) Schematic diagram of reflective metasurface hologram with LCP and RCP; (e) Simulated results with different GI algorithms; (f) Experimental results with different GI algorithms; (g) Scheme of the optical encryption based on CGI with helicity-dependent metasurface hologram [65]
图 6 (a)一维超材料孔径实现压缩成像的结构示意图;(b)不同静态目标的重构图像[77];(c)采用动态超表面天线的成像系统;(d)动态超表面天线和激励信号示意图[79]
Figure 6. (a) Structural diagram of 1D resonant metamaterial apertures for compressed imaging; (b) Reconstructions of different static scenes[77]; (c) Imaging system consisting of dynamic metasurface antennas; (d) Schematic diagram of the dynamic metasurface antennas and the exciting signals[79]
图 7 (a)利用DMA的无相计算微波鬼成像配置图;(b)由DMA产生的结构化照明掩模图像[81];(c)实现非相干压缩鬼成像的DMA示意图;(d)实验测得的通过平均左边的接收机和右边的发射机的调谐状态恢复的两个目标图像[82]
Figure 7. (a) Scheme of phaseless computational ghost imaging at microwave frequencies using DMA; (b) Structured illumination patterns generated by a DMA [81]; (c) Schematic illustration of DMA to achieve incoherent CGI; (d) Reciprocal images of two targets obtained by averaging over receiver tuning states on the left, transmitter tuning states on the right, and the product of them[82]
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