⇦Chinese Physics Letters, 2019, Vol. 36, No. 11, Article code 114203 Label-Free Microscopic Imaging Based on the Random Matrix Theory in Wavefront Shaping * Li-Qi Yu (俞力奇)1, Xin-Yu Xu (徐新羽)1, Zhen-Feng Zhang (张振峰)1, Qi Feng (冯祺)1, Bin Zhang (张彬)2, Ying-Chun Ding (丁迎春)1**, Qiang Liu (柳强)2** Affiliations 1College of Mathematics and Physics, Beijing University of Chemical Technology, Beijing 100029 2State Key Laboratory of Precision Measurement Technology and Instruments, Department of Precision Instruments, Tsinghua University, Beijing 100084 Received 23 August 2019, online 21 October 2019 *Supported by the National Key Research and Development Program of China under Grant No 2017YFB1104500, the Beijing Natural Science Foundation under Grant No 7182091, the National Natural Science Foundation of China under Grant No 21627813, and the Research Projects on Biomedical Transformation of China-Japan Friendship Hospital under Grant No PYBZ1801.
**Corresponding author. Email: dingyc@mail.buct.edu.cn; qiangliu@mail.tsinghua.edu.cn Citation Text: Yu L Q, Xu X Y, Zhang Z F, Feng Q and Zhang B et al 2019 Chin. Phys. Lett. 36 114203    Abstract Wavefront shaping technology has mainly been applied to microscopic fluorescence imaging through turbid media, with the advantages of high resolution and imaging depth beyond the ballistic regime. However, fluorescence needs to be introduced extrinsically and the field of view is limited by memory effects. Here we propose a new method for microscopic imaging light transmission through turbid media, which has the advantages of label-free and discretional field of view size, based on transmission-matrix-based wavefront shaping and the random matrix theory. We also verify that a target of absorber behind the strong scattering media can be imaged with high resolution in the experiment. Our method opens a new avenue for the research and application of wavefront shaping. DOI:10.1088/0256-307X/36/11/114203 PACS:42.30.-d, 42.25.-p, 42.25.Dd, 42.25.Fx © 2019 Chinese Physics Society Article Text Wavefront shaping technology has many potential widespread applications, due to its ability to control light propagation through turbid media, such as optical trapping of particles,[1] astronomical imaging,[2] spectral filtering,[3] quantum authentication[4] and communication.[5,6] In particular, it is often used for microscopic imaging through turbid biological tissues,[7–15] because the imaging depth can exceed the ballistic light regime.[16] At present, wavefront shaping technology is mainly divided into three categories: transmission matrix method,[7,8] optical phase conjugation method and iterative optimization method. Popoff et al. reconstructed the image displayed on a spatial light modulator (SLM) from the speckle on the CCD camera by measuring the transmission matrix of the turbid medium.[7] However, few researchers have used the transmission matrix method to microscopic image through turbid media. The time-reversed ultrasound-encoded method (TRUE) is used to focus light inside turbid media to exciting the fluorescence on the imaged target, which has a fast imaging speed.[9,10] Some iterative optimization algorithms are used to control the wavefront of the light to focus the light on the target after the turbid medium, and use the memory effect (ME) to scan the target imaging behind the turbid medium.[12–15] The imaging resolution of the iterative optimization method can break through the diffraction limit.[17] However, these methods also have some limitations for light microscopic imaging through turbid biological tissues. First, the imaging resolution of the TRUE method is limited by the size of the ultrasound focus,[18] and the imaging system is not robust. Second, fluorescence needs to be introduced from extrinsically, either invasively or by physical contact,[19] and it may be harmful to humans. Furthermore, autofluorescence of the surrounding tissues poses a noticeable background and seriously impairs the quality of image. Finally, the field of view (FOV) of an image is intrinsically limited by the ME range, which is inversely proportional to the thickness of the turbid media. In other words, the thicker the turbid media, the smaller the imaging range. Furthermore, the FOV is 0 when the target is attached to the turbid media. In the experiment of memory effect, the imaging target needs to have a certain distance from the turbid medium. This imaging structure is not suitable for using in biomicroscopic imaging field.[20] Moreover, the intensity of light is not uniform in the scanning range of ME. In this Letter, we introduce a new imaging method that the target is label-free and the imaging has discretional FOV size, based on transmission-matrix-based wavefront shaping and the random matrix theory.[21] In our experiment, we use co-propagation interference to measure the transmission matrix,[22] and we use our method to image a pink absorber 'F' hidden behind a strong scattering media. Our imaging method also has a high resolution. The imaging resolution is higher than 20 µm and the FOV is 150$\times$150 µm$^{2}$. In our method, the FOV can be of any size, only related to the input and output field sizes selected. The higher resolution can be achieved when choosing more modes or a smaller FOV. The Pearson correlation coefficient is used to evaluate the imaging quality, and the Pearson correlation coefficient between the reconstructed image and the original image reaches 0.82 after de-noising. Using inherent difference of light absorption between different biological tissues, we will be able to image different biological tissues through turbid media, such as nucleus and erythrocyte. Depending on the spectral absorption characteristics of different biological tissues, we will also be able to choose different light wavelengths as a light source. Consequently, our method holds great potential for application in endogenous chromogenic imaging in deep biological tissues.
cpl-36-11-114203-fig1.png
Fig. 1. (a) The absorber F in the optical path. (b) The diagonal matrix $P$ represents the effect of light passing through the absorber F on the measured transmission matrix. (c) An absorber behind the turbid media.
If light passes through an absorber in the process of measuring the transmission matrix of the turbid medium, then the measurement of the transmission matrix of the turbid medium will be affected. In fact, the transmission matrix that we measured contains not only the transmission matrix of the turbid medium but also the transmission matrix of the absorber. In the process of measuring the transmission matrix, we assume that there is an absorber F in the optical path. As shown in Fig. 1, the effect of light passing through the absorber F on the measured transmission matrix can be represented by an $N\times N$ matrix $P$, which is a diagonal matrix whose diagonal elements represent the ability to absorb light at each position of the absorber. Among the diagonal elements, the element value at a certain position where there is no light absorption is 1, and the stronger the light absorption ability is, the smaller the corresponding element value is (closer to 0). For the case of Fig. 1(c), the following is a detailed process of reconstructing the image of the absorber behind the turbid medium. First, the $N\times N$ transmission matrix ${\boldsymbol K}_{\rm obs}$ that we measured by mean of co-propagation interference method[22] is $$\begin{align} {\boldsymbol K}_{\rm obs}=PK,~~ \tag {1} \end{align} $$ where $K$ is the transmission matrix of the turbid media, and $P$ is the diagonal matrix of the absorber. Equation (1) can also be written as $$\begin{align} {\boldsymbol K}_{\rm obs}=\,&\left[\begin{matrix} p_{1} & 0&\cdots & 0\\ 0 & p_{2}&\cdots & 0\\ \vdots & \vdots& \ddots & \vdots \\ 0 & 0&\cdots & p_{N}\\ \end{matrix}\right]\\ &\cdot\left[\begin{matrix} k_{11} & k_{12}&\cdots & k_{1N}\\ k_{21} & k_{22}&\cdots & k_{2N}\\ \vdots & \vdots&\ddots & \vdots \\ k_{N1} & k_{N2}&\cdots & k_{NN}\\ \end{matrix}\right]\\ =\,&\left[\begin{matrix} {p_{1}k}_{11} & p_{1}k_{12}&\cdots & p_{1}k_{1N}\\ {p_{2}k}_{21} & {p_{2}k}_{22}&\cdots & {p_{2}k}_{2N}\\ \vdots & \vdots &\ddots & \vdots \\ {p_{N}k}_{N1} & {p_{N}k}_{N2}&\cdots & {p_{N}k}_{NN}\\ \end{matrix}\right]\\ =\,&\left[\begin{matrix} k_{11}^{{\rm obs}} & k_{12}^{{\rm obs}}&\cdots & k_{1N}^{{\rm obs}}\\ k_{21}^{{\rm obs}} & k_{22}^{{\rm obs}}&\cdots & k_{2N}^{{\rm obs}}\\ \vdots & \vdots &\ddots & \vdots \\ k_{N1}^{{\rm obs}} & k_{N2}^{{\rm obs}}&\cdots & k_{NN}^{{\rm obs}}\\ \end{matrix}\right].~~ \tag {2} \end{align} $$ Second, according to the random matrix theory,[21] each element in the transmission matrix $K$ follows the circle Gaussian random distribution with a mean of 0. Thus statistically, for any row elements or column elements of $K$, there exists $$\begin{align} &|k_{11}|+|k_{12}|+\cdots +|k_{1N}|\\ \approx\,&|k_{n1}|+|k_{n2}|+\cdots +|k_{nN}|\\ \approx\,&a~({\rm constant}).~~ \tag {3} \end{align} $$ From Eqs. (2) and (3), we can obtain $P$ $$\begin{alignat}{1} P=\,&{\rm diag}(p_{1} p_{2} \cdots p_{N})\\ \approx\,&{\rm diag}(p_{1}(|k_{11}|+|k_{12}|+\cdots +|k_{1N}|)\\ &p_{2}(|k_{21}|+|k_{22}|+\cdots +|k_{2N}|)\\ &\cdots p_{N}(| k_{N1}|+|k_{N2}|+\cdots +|k_{NN}|))/a\\ =\,&{\rm diag}((|{p_{1}k}_{11}|+|p_{1}k_{12}|+\cdots +|{p_{1}k}_{1N}|)\\ &(|p_{2}k_{21}|+|{p_{2}k}_{22}|+\cdots +|p_{2}k_{2N}|)\\ &\cdots (|p_{N}k_{N1}|+|p_{N}k_{N2}|+\cdots +| p_{N}k_{NN}|))/a\\ =\,&{\rm diag}\Big(\sum\limits_j |k_{1j}^{{\rm obs}}| \sum\limits_j |k_{2j}^{{\rm obs}}| \cdots \sum\limits_j | k_{Nj}^{{\rm obs}}| \Big)/a.~~ \tag {4} \end{alignat} $$ Finally, according to Eq. (4), we can reconstruct the image of the absorber from the diagonal matrix $P$. That is, if we measure the value of ${\boldsymbol K}_{\rm obs}$ in the experiments, we can reconstruct the image of the absorber by solving Eq. (4). We have carried out experiments in the situation shown in Fig. 1(c). The experimental configuration is shown in Fig. 2(a). The light emitted from the laser is modulated by the digital micromirror device (DMD) and propagates along the optical path, and finally received by the CCD camera. Here L1, L2 and SF form a 4$f$ system for filtering out other diffraction orders except $+$1 order in the superpixel method.[23] The sample we made is a mixture of ZnO and varnish as strong scattering media on one side of the slide glass to simulate turbid biological tissue. On the other side of the slide glass, we made an absorber F with pink dye, which can absorb 532 nm light. The physical sample of the scattering sample is shown in Figs. 2(b)–2(d). Here Fig. 2(d) is a photograph taken by a CCD camera using a microscope. The overall size of the pattern is $150\times150$ µm$^{2}$.
cpl-36-11-114203-fig2.png
Fig. 2. (a) The configuration of experiment. Laser: 532 nm laser source, EL: beam expander, P1, P2: polarizer, HWP: half-wave plate, DMD: digital micromirror device, L1, L2: lens, SF: spatial filter, S: sample, obj1, obj2: 20$\times$ objective lens, CCD: CCD camera. (b) The ZnO layer. (c) The pattern layer. (d) The absorber F.
In the experiment, we measured the transmission matrix with a co-propagation interference method,[22] thus an additional reference arm was not needed. In addition, we used a digital micromirror device (DMD) combined with the superpixel method instead of using an SLM to modulate phases of incident light.[23,24] The DMD (Texas Instruments, DLP 6500, 1920$\times$1080 pixels) we used here has an extremely fast refresh rate up to 23 kHz. The adjacent 4$\times$4 micromirrors are grouped into single superpixel and each superpixel defines a complex field in the target plane. After a plane light beam illuminates the DMD surface with an angle, the reflected light is divided into many diffraction orders. In the $+$1 or $-$1 order, the phase is modulated by the superpixel-based DMD.[8] However, limited by the CCD's refresh rate, the actual speed in the experiment is 5 Hz, and the total number of loaded masks is 3096. As a result, the modulation time is 619.2 s. The experimental results are shown in Fig. 3. In Fig. 3(a), the neighbor interpolation is used to reduce Fig. 2(d) to 32$\times$32 pixels as the original image. Figure 3(b) is the speckle pattern taken by a CCD camera when unmodulated light passes through turbid media. We can see that the image of the target is completely submerged in the speckle pattern. After measuring the transmission matrix ${\boldsymbol K}_{\rm obs}$, the image was reconstructed by Eq. (4). To improve the signal-to-noise ratio, the local mean square deviation de-noising and wavelet de-noising were used. We calculated the Pearson correlation coefficient of the image to represent the quality of the reconstructed image. The reconstructed image of the target after de-noising is shown in Fig. 3(c) and the Pearson correlation coefficient between the reconstructed image and the original image reaches 0.82. In this work, we have not studied the limitation of imaging resolution, only a lower limit of resolution is given according to the target size of our sample. From Fig. 3(c), we can see that the imaging resolution is greater than 20 µm. As a comparison, we researched the wavefront shaping imaging method based on the traditional transmission matrix. We used the scanning method to image the absorber behind the scattering media. First, using the measured transmission matrix ${\boldsymbol K}_{\rm obs}$, light was focused on each position of the absorber behind the scattering media in turn. Then, the image of the absorber was reconstructed by detecting the intensity at each position. The result is shown in Fig. 3(d), which was also processed by the same de-noising algorithm, and the Pearson correlation coefficient is only 0.78. The experimental results show that our method is more efficient than the scanning method in reconstructing images through scattering media. This is because our method avoids the problem of uneven focus points in the scanning method. In addition, our method is less influenced by noise interference than the scanning method, effectively avoiding the noise generated during scanning.
cpl-36-11-114203-fig3.png
Fig. 3. (a) The original image. (b) The speckle pattern of light passing through the turbid media. (c) The image reconstructed by our method. (d) The image reconstructed by scanning method.
In summary, we have introduced the principle of our imaging method and the reconstruction process of the target object behind the turbid medium in detail. Wavefront shaping based on random matrix theory is one of the techniques of scatter imaging. It can recover the target information by 'decoding' the messy information from the chaotic image that cannot distinguish the original information due to scattering. We have verified in the experiment and simulation that our method can image the absorber behind the turbid media with high resolution. Compared with some methods of fluorescence imaging in wavefront shaping, our method holds the advantages of label-free and non-invasiveness. Moreover, we can choose a discretionally sized FOV in our method that is more flexible than the memory effect scanning. The imaging target can also be attached to the turbid medium. Therefore, our method has great potential for application in the field of microscopic imaging of light transmitting through turbid biological tissue. Moreover, we use the scanning method to image the absorber behind the scattering media as a comparison. As can be seen from the results, our method is better than the scanning method in reconstructing images through scattering media. According to our analysis, there are two reasons for this: (1) the scanning method can not make each position have the same focus intensity, and different focus intensities will increase noise; and (2) environmental noise will be introduced during the scanning progress of the focus point. In contrast from the scanning method, our method does not require scanning, thus the influence of these two kinds of noises can be avoided. However, our method is currently limited by the hardware speed in measuring the transmission matrix and cannot be applied to image living tissue. In the future, we will continue to improve our method and hope to achieve a better biomedical imaging result under a better hardware condition. References In situ wavefront correction and its application to micromanipulationSpectral control of broadband light through random media by wavefront shapingContinuous-variable quantum authentication of physical unclonable keysTaking Advantage of Multiple Scattering to Communicate with Time-Reversal AntennasFocusing Beyond the Diffraction Limit with Far-Field Time ReversalImage transmission through an opaque materialRapid measurement of transmission matrix with the sequential semi-definite programming methodDeep-tissue focal fluorescence imaging with digitally time-reversed ultrasound-encoded lightFluorescence imaging beyond the ballistic regime by ultrasound-pulse-guided digital phase conjugationTime-reversed adapted-perturbation (TRAP) optical focusing onto dynamic objects inside scattering mediaScattered light fluorescence microscopy: imaging through turbid layersIn vivo volumetric imaging of biological dynamics in deep tissue via wavefront engineeringResearch on intelligent algorithms for amplitude optimization of wavefront shapingShaping the Wavefront of Incident Light with a Strong Robustness Particle Swarm Optimization AlgorithmGuidestar-assisted wavefront-shaping methods for focusing light into biological tissueScattering Lens Resolves Sub-100 nm Structures with Visible LightIterative Time-Reversed Ultrasonically Encoded Light Focusing in Backscattering ModeThrough-skull fluorescence imaging of the brain in a new near-infrared windowLooking around corners and through thin turbid layers in real time with scattered incoherent lightLight fields in complex media: Mesoscopic scattering meets wave controlControlling light through optical disordered media: transmission matrix approachSuperpixel-based spatial amplitude and phase modulation using a digital micromirror deviceSuperpixel-Based Complex Field Modulation Using a Digital Micromirror Device for Focusing Light through Scattering Media
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