Line-scanning confocal microscopic imaging based on virtual structured modulation

ZHAO Jia-wang，ZHANG Yun-hai ，WANG Fa-min，MIAO Xin，SHI Xin

（1.School of Biomedical Engineering, University of Science and Technology of China, Hefei 230026, China；2.Jiangsu Key Lab of Medical Optics, Suzhou Institute of Biomedical Engineering and Technology,Chinese Academy of Sciences, Suzhou 215163, China；3.The Second Affiliated Hospital of Soochow University, Suzhou 215000, China）

Abstract:Resolution in a confocal microscope is limited by the diffraction limit.Structured modulation has been proven to be able to achieve super-resolution in confocal microscopy,however,its limited speed in image acquisition limits its applicability in practical applications.In order to improve its imaging speed,we introduce a method that can achieve rapid super-resolution confocal microscopy by combining line-scanning and structured detection.A cylindrical lens is used to focus the light into a line,and a digital mask with a sinusoidal function is used to modulate the descanned image in the light detection arm.Unlike the virtual structured method,there is no need for a subsequent frequency shift process.In order to improve the isotropic resolution of the system,a scanning angle of 0°and 90°is achieved by rotating the sample.Simulation and experiment results indicate that the spectrum width of coherent transfer function expands and the resolution is 1.4 times as large as that of a conventional confocal microscope.This method increases the system’s imaging acquisition speed 104-fold when compared with a confocal structured modulation microscope that uses spotscanning.

Key words:line-scanning confocal;super-resolution;virtual structured modulation

1 Introduction

Due to the diffraction effect of light wave,the resolution of a traditional optical microscope is limited[1-2].Laser-Scanning Confocal Microscopy(LSCM)has a higher resolution than wide-field microscopy because it uses a tightly focused excitation beam and pinhole detection to suppress the defocusing background light[3-4].However,due to the limitation of pinhole size,the resolution of a confocal microscope with smaller pinhole is achieved at the price of SNR reduction.To achieve their balance,the pinhole size is generally large,resulting in a lateral resolution lower than the ideal result but still within the diffraction limit[5-6].In the past two decades,many super-resolution optical microscopy methods such as Stimulated Emission Depletion(STED)microscopy[7]and Structured Illumination Microscopy(SIM)[8]have been applied.These methods follow two main principles,namely,decreasing the size of Point Spread Function(PSF)and increasing the bandwidth of Optical Transfer Function(OTF)[9].In addition,Stochastic Optical Reconstruction Microscopy(STORM)[10]and Photo-Activated Localization Microscopy(PALM)[11]can achieve super-resolution by using an optically switched fluorescent probe to locate a single molecule.These methods have made breakthroughs in fluorescencelabeled imaging resolution.However,each of them also has some limitations.PALM and STORM have long been limited by their imaging speed.In the STED microscopy,the excitation and emission spectra are required to match the given excitation wavelength and depletion wavelength.SIM can only image optical thin samples[12].

The structural detection microscope is derived from the principle of structured illumination and its resolution enhancement concept is similar to Moire fringe.By acquiring images through optical masks at different scanning locations,the OTF bandwidth is doubled compared with a traditional microscope[13].However,wide-field space structure illumination is not suitable for a scanning microscope,due to the need for a patterned mask(such as a grating)at the illumination end.With the proposing of Scanning Patterned Illumination(SPIN)microscopy and Scanning Patterned Detection(SPADE)microscopy,super-resolution laser scanning microscope has been realized.These methods have achieved the same effect as SIM in the point scanning system by using the time and space modulation[14].SPADE,also known as Virtual Structure Detection(VSD),has been proven correct.However,since the spot image of each scanning point is needed,the imaging speed in VSD is severely limited[15].

To overcome the above shortcomings,virtual structured modulation has been applied to largeaperture confocal microscope by means of Linescanning confocal microscopy with Virtually Structured Modulation(LVSM)microscopy[16],thus improving the speed and resolution of confocal microscopic imaging.Its difference from VSD is that no subsequent frequency shift is required.At the same time,thick samples can be imaged due to the unique slicing ability of confocal microscope.Different from most of the other super-resolution imaging methods,this method can image non-fluorescent samples with high resolution and fast imaging speed.

2 Principle of line-scanning confocal virtual structure modulation

The reflective confocal coherent imaging system is shown in Fig.1.

Fig.1 Schematic of reflective confocal microscope system图1 反射式共聚焦系统示意图

Suppose the illumination intensity and the system amplification factor are 1.In the case of unscanning,the phase factor is ignored after the interaction between the illumination light field and the sample.Then the amplitude distribution on the sample surface will be:

wherehil(x,y)is the two-dimensional amplitude PSF of the illumination path,sis the amplitude distribution of the sample,and(x1,y1)is the location of the sample.After passing through the detection light path,the contribution of the light field amplitude distributionA1(x,y,x1,y1)obtained after the interaction between the sample and the illumination light field to the amplitude of a point(x2,y2)on the image plane can be expressed as:

wherehde(x,y)is the two-dimensional amplitude PSF of the detection path,andA2(x,y,x1,y1,x2,y2)is the contribution of the amplitude distribution obtained after the interaction between the sample point(x1,y1)and the illumination light field to the point(x2,y2)on the image plane.The superposition of the contributions of all the sample points to the amplitude at that point is just the amplitude detected on the image plane.The amplitude distribution on the whole image plane is:

whereA3(x1,y1,i,j)is the amplitude distribution on the image plane.If the detection function isD(i,j),then:

whereA(x1,y1)is the amplitude image finally detected by the detector.Bothhil(x,y)andhde(x,y)are even functions.The definition of convolution is applied to obtain:

It can be seen from Eq.(5)that the final amplitude image is the superposition of the amplitudes at different positions of the sample.This is a coherent imaging process.The imaging characteristics of the system depend on the amplitude PSF:

Then the Coherent Transfer Function(CTF)of the system can be expressed as the Fourier transform of APSF:

whereHil(fx1,fy1)andHde(fx1,fy1)are the CTFs at the illumination end and detection end respectively,⊗represents two-dimensional convolution,and F is the symbol of Fourier transform.According to Eq.(6),the Intensity Point Spread Function(IPSF)of the system can be expressed as[17]:

Since the imaging performance of the system is ultimately limited by lens,lighting mode and detection function,the forms of system amplitude PSF under different conditions will be discussed separately.

The scanning mode in which the lighting mode is point lighting andD(x1,y1)is a virtual pinhole is point-scanning mode.The amplitude PSFs for illumination path and detection path can be derived from the Fourier transform of the lens aperture.The effective detection PSF is the convolution of the detection PSF and the aperture functionD(x1,y1).

The scanning mode in which the lighting mode is line lighting andD(x1,y1)is a virtual slit is linescanning mode.Compared with point scanning,the amplitude PSF of the detection path remains unchanged.Since a cylindrical lens is introduced to focus the spot on the illumination path,the amplitude PSF of the illumination path will change into Gaussian distribution in one direction and constant distribution in the other direction.Suppose the scanning direction is along theXaxis,that is,the illumination PSF is a constant distribution in theYaxis direction.Then under the paraxial approximation,the amplitude PSF of the illumination path can be written into[18]:

whereΦ=2πNA/λ,w≪1.Therefore,the first term in Eq.(9)can be ignored.This means that the amplitude PSF of the illumination path has a constant excitation along the linear direction.

The LVSM method proposed in this paper is just the superposition of cosine mask and line scanning(line lighting,andD(x1,y1)=rectangular slit).In this method,the amplitude PSFs for illumination path and detection path are the same as those in linescanning mode,but the detection functionD(x1,y1)changes into:

wherepis the slit width;sis the length of line spot on the CCD,and is infinitely small when being analyzed in the above line-scanning PSF model.Based on the line-scanning mode,this paper mainly studies the characteristics of coherent imaging in the system when the detection function is as shown in Eq.(10).Since only the super-resolution information in the current scanning direction can be extracted under the single-line-scanning condition,the Eq.(5)can be rewritten into(when only one dimension,namelyx-axis,is considered):

By substituting the Eq.(10)into Eq.(11),the Fourier transform of Eq.(11)will be

whereUc(fx1)is low frequency component,andUl(fx1)andUr(fx1)are high frequency components.They can be easily determined from Eq.(13).

3 Computational simulation

The laser wavelength λof the simulated lighting is 488 nm,the numerical aperture of the objective lens is 0.4,the cosine function modulation frequencyf0is 0.9NA/λ,and both θand φare 0.The scanning direction isXdirection,i.e.,0°direction.The extreme case is assumed,i.e.the slit widthsis infinitely small.The IPSF simulating the normal line-scanning confocal imaging in accordance with Eq.(8)is shown in Fig.2(Color online).According to Fig.2(c),the Full Width at Half Maximum(FWHM)in theXdirection decreases to 0.71 times of the FWHM in theYdirection,because of the confocal imaging in theXdirection and the wide-field imaging in theYdirection.It can also be seen from the OTF in Fig.2(b)that the cut-off frequency in theXdirection is higher than that in theYdirection,so the ability to transmit high-frequency information is enhanced.

Fig.2 Theoretical simulation results of IPSF.(a)IPSF of traditional line-scanning confocal microscope;(b)Fourier transform of IPSF;(c)normalized intensity distributions of(a)in theXandYdirections respectively图2 IPSF 理论仿真结果。(a)普通线扫描共聚焦显微镜的IPSF，(b)IPSF 的傅立叶变换，(c)图(a)中X 和Y 方向上的归一化强度分布

The simulation system CTF based on Eq.(7)is shown in Fig.3(Color online),where the slit width is equal to an Airy disk.It can be seen from Fig.3(c)that,compared with Wide-field Microscopy(WM),the cut-off frequencies of CLSM and LVSM in the direction of image scanning are both twice that of WM,indicating higher resolution in the corresponding spatial domain.Given the same cut-off frequency,the percentage of high-frequency information in LVSM is significantly higher than that in CLSM.This indicates that LVSM has a stronger capability of high-frequency information transfer.This is attributed to the modulation of cosine function,which increases the high-frequency ratio.

In the structure detection method based on linescanning confocal imaging,the structure detection functionD(x,y)directly acts on the unscanned image of the detector plane and doesn’t need to be conjugated with the sample,and the detection function image remains unchanged.The difference between this method and VSD super-resolution method is that,in the VSD method,the structure detection function acts on non-unscanned images,but the imaging system itself is unscanned,so the structure function images need to be conjugated with the sample in order to achieve the effect of non-unscanning modulation and obtain the image data similar to wide-field SIM.In VSD,the reconstruction algorithm identical with wide-field SIM is used to move high-frequency information to the right place,in order to achieve the super-resolution effect.However,in the above method,the unscanned images under line-scanning confocal condition are directly modulated by the detection function,and the cut-off frequency of the system CTF is extended without frequency shift.Even without subsequent image processing,the image resolution is still improved.However,the problem of low transmittance of high-frequency information,as shown in the red curve in Fig.3(c),still exists,so the improvement effect is not obvious.As shown in Fig.4,we can change the phases of the modulation function,establish an equation like Eq.(13),calculate the frequency components along these directions,use the Wiener filtering algorithm to reduce noise,and then apply the generalized Wiener filter to perform weighted superposition recovery of the processed frequency domain information in order to enhance the high frequency components.When the generalized Wiener filter is used for weighted superposition,the weight factor of each frequency component mainly depends on its SNR,which can be estimated with the method in Ref.[19].After the frequency domain image weighted and superimposed enters the spatial domain through inverse Fourier transform,the finally reconstructed amplitude image can be obtained.Then through square operation,the intensity image can be obtained.

Since line-scanning confocal imaging is working only in one direction,the specific detection function can only expand the cut-off frequency of CTF in a single direction of the system.In order to illustrate the feasibility of isotropic resolution improvement,the image field needs to be rotated to obtain the line-scanning data in different directions for virtual modulation.The more the modulation directions are,the more obvious the isotropic resolution improvement will be.Meanwhile,the imaging speed rate will be reduced,but still much higher than that in the point-scanning mode.An appropriate modulation direction can be selected according to the actual situation.The line spot image obtained in each scanning position is modulated by the structure detection function,and the integral of the images within the linear spot in one direction is calculated to obtain a series of values representing the current scanning positions.These values and their positions represent the structurally modulated image and are used in the subsequent reconstruction process.Because the modulation mode is virtual modulation and the direction and phase of the digital mask can be accurately known,the estimated error of modulation mode and phase will not exist.

Fig.3 System CTF simulation.(a)CTF of traditional confocal line scanning microscopy(for CLSM);(b)CTF of line-scanning confocal microscopy with structure modulation(for LVSM);(c)black curve is the normalized frequency distribution along theYdirection in(a)(for WM),and blue and red curves are the normalized frequency distributions along theXdirection in(a)and(b),respectively(for CLSM and LVSM)图3 系统CTF 仿真。(a)传统线扫描共聚焦(CLSM)的CTF，(b)线扫描结构调制共聚焦(LVSM)的CTF，(c)黑色曲线为(a)中Y 方向归一化频率分布(WM)，蓝色和红色曲线分别为(a)、(b)中X 方向归一化频率分布(CLSM、LVSM)

Fig.4 Flow chart of image reconstruction图4 图像重建过程流程图

To demonstrate the effectiveness of the above method,the imaging results of LVSM were simulated,as shown in Fig.5(Color online).The scanning directions of each sample were 0°and 90°.It can be seen from Fig.5(e)and Fig.5(f)that part of the high-frequency information is cut off,and the detail information of the sample is lost.After the LVSM reconstruction,the high-frequency information in the corresponding direction is moved into the frequency domain passband of the system and put in the right place,and the frequency spectrum in the scanning direction is expanded.It can be seen from the comparison between Fig.5(b)and Fig.5(c)that,after an image in the frequency domain is moved into the spatial domain through transformation,its resolution is significantly improved.

Fig.5 Simulation of line-scanning confocal microscopy with virtual structure modulation(for LVSM).(a)Spoke-like sample for simulation;(b)image of conventional confocal microscopy;(c)image reconstructed with the structure detection functions in the two scanning directions of 0°and 90°;(d),(e)and(f)are the Fourier transforms i.e.frequency domain images of(a),(b)and(c),respectively图5 线扫描共聚焦虚拟结构调制仿真（LVSM）。(a)仿真使用的辐条状样品。(b)普通共聚焦图像。(c)取0°、90°两个扫描方向，结合对应方向上的结构检测函数重建后图像。(d)、(e)、(f)分别是(a)、(b)、(c)的傅立叶变换，即对应的频域图像

4 Experiments and results

4.1 Experimental system

The LVSM based on laser line-scanning structure detection is shown in Fig.6.In the LVSM,a single-mode He-Ne laser with a wavelength of 633 nm is used to generate the polarized laser,and an optical attenuation is used to reduce the light intensity.The cylindrical lens CL(f=180 mm)has the focusing characteristic only in one direction,focusing the attenuated laser into a line of light that will be incident to a uniaxial scanning galvanometer(Mode 6215 CTI).The scanning galvanometer vibrates in the scanning direction to guide the focusing line through the homemade scanning lens,tube lens(TTL 180-A Olympus)and objective lens until the line moves along the specified direction on the sample.To reduce the vignetting effect,the center of the galvanometer is controlled to conjugate with the pupil plane of the objective lens.The light reflected from the sample is unscanned by one-dimensional scanning system and relayed to the image plane via the intermediate optical system.The detector collects the line spot images from the current scanning position.As the scanning galvanometer swings,the position of the unscanned image on the detector plane remains unchanged but the image information is constantly updated.512 line-spot images are continuously collected from different scanning positions to obtain a two-dimensional image.

Fig.6 Schematic diagram of experiment setup图6 实验系统示意图

sCMOS,a two-dimensional image acquisition device,is used to acquire a single line-spot image(ORCA Flash4.0 V2,Hamamatsu).Compared with EMCCD,the sCMOS has higher quantum efficiency and lower noise output.When the ROI of line-spot image is set as 512 pixel×64 pixel,the theoretical image acquisition rate in a single direction can reach 3 206 fps.In the actual experiment,considering the response time of the device and the delay of the program,the image acquisition rate will decrease,and the image acquisition time in a single direction will be about 0.25s.The width of virtual slit is assumed to be the diameter of Airy disk and is used to detect and process the collected images.

A 4×flat-field semiapochromat(Olympus)with a numerical aperture of 0.13 is used.The experimental sample is a target with standard optical resolution(USAF 19511×,Edmund).To verify the enhancement of final isotropic resolution,an electric rotary translation stage with the maximum rotation rate of 50°/s is driven by a stepper motor to rotate the sample.The directions of image acquisition are 0°and 90°,and the time for the whole image rotation process is about 2s.

4.2 Experimental results

Some of the acquired line-spot images and the final image reconstruction results are shown in Fig.7(Color online).

Fig.7 Implementation of line-scanning confocal virtual structure modulation imaging on the resolution test target.(a)The 20th,205th,360th and 490th line spot images collected in the 0°scanning direction;(b)the 20th,205th,360th and 490th line spot images collected in the 90°scanning direction;(c)the image of resolution test target obtained by conventional line-scanning confocal method in the 90°scanning direction;(d)reconstructed super-resolution image by LVSM图7 分辨率测试目标的线扫描虚拟结构调制共聚焦实现。(a)扫描方向为0°时，采集的第20、205、360、490 条线斑图像。(b)扫描方向为90°时，采集的第20、205、360、490 条线斑图像。(c)扫描方向为90°时，常规线扫描共聚焦获得的分辨率测试靶图片。(d)LVSM 超分辨重建后图像

The areas marked by blue line segments are the No.8.2～8.6 line pairs in theYdirection of the resolution test board,the area marked by yellow line segment is the No.8.5 line pair in theXdirection,and the area marked by green line segment is the No.8.6 line pair.It can be seen from Fig.8(a)(Color online)that in theYdirection,a conventional line-scanning confocal microscope can distinguish the No.8.2 group but cannot distinguish the No.8.3 group.The number of line pairs in the No.8.2 group is 287 lp/mm,and the corresponding spatial period is 3.48μm.The LVSM microscope can distinguish the No.8.5 group but cannot distinguish the No.8.6 group.The number of line pairs in the No.8.5 group is 406 lp/mm,and the corresponding spatial period is 2.46μm.The resolution of LVSM microscope is higher than that of a line-scanning confocal microscope with the same slit size.As seen from Fig.8(b)(Color online)and Fig.8(c)(Color online),in theXdirection,the line-scanning confocal microscope cannot distinguish both the No.8.5 and No.8.6 groups.In comparison,the LVSM can distinguish the No.8.5 group but cannot distinguish the No.8.6 group.Therefore,the resolutions of LVSM in both theXandYdirections are improved to 2.46μm,1.4 times that of traditional line-scanning confocal microscope.

Fig.8 Normalized intensity curves of conventional line-scanning confocal microscope and line-scanning confocal microscope with virtual structure modulation in specified areas.Comparison between the normalized intensities of the area marked by(a)blue curve,(b)yellow curve and(c)green curve in Fig.7(c)and Fig.7(d)图8 常规线扫描共聚焦和线扫描虚拟结构调制共聚焦在特定区域的归一化强度曲线。图7(c)、7(d)中(a)蓝色线段标记区域(b)黄色线段标记区域及(c)绿色线段标记区域的归一化强度分布对比

The above experiment demonstrates that the LVSM can break the diffraction limit during highspeed imaging.This theory shows that the LVSM can improve the lateral resolution and imaging speed by using the modulation factors in two directions.The experiment with standard resolution target confirms that in addition to performing the line scanning based on high-speed imaging,the LVSM microscope can show the detailed target structure that can’t be detected by conventional confocal microscopy.

5 Conclusion

In this paper,a line-scanning confocal microscopic imaging method based on structural modulation is presented.The related theories and reconstruction methods are deduced and verified by experiment.The results of simulation and experiment show that the system CTF is enlarged and the imaging resolution is 1.4 times that of traditional confocal microscope.Compared with the point-scanning spot imaging with virtual structure modulation,the imaging rate of the system can be greatly improved in this method.This method needs 2.5 s to scan the image with 512 pixel×512 pixel in two directions,104 times faster than the former method,which needs about 260 s to complete the image acquisition under the image field of the same size.

——中文对照版——

1 引言

由于光波衍射效应的存在，传统光学显微镜的分辨率受到限制[1-2]。激光扫描共焦显微镜（LSCM）具有比宽场显微镜更高的分辨率。因为它使用紧密聚焦的激发光束和针孔检测来抑制离焦背景光[3-4]。但是受到针孔大小的限制，在共焦显微镜中，减小针孔的尺寸在提高分辨率的同时也降低了信噪比。为了保持二者的平衡，针孔尺寸一般较大，这会导致横向分辨率低于理想结果，仍处于衍射极限之内[5-6]。近20 年来，许多超分辨光学显微方法得到了推广，如受激发射损耗显微镜（STED）[7]、结构光照明显微镜（SIM）[8]等，这些方法主要基于两个原理，分别是压缩点扩散函数（PSF）和增加光传递函数（OTF）带宽[9]。除此之外，随机光学重建显微镜（STORM）[10]和光激活定位显微镜（PALM）[11]通过使用光开关荧光探针定位单个分子来实现超分辨率。上述这些方法在荧光标记成像分辨率上都有突破，但也都有其局限性。PALM 和STORM长期以来一直受到成像速度的限制，STED 显微镜要求激发光谱和发射光谱必须与给定的激发和耗尽波长相匹配，SIM 只能成像光学薄样品[12]。

结构探测显微镜基于结构照明原理，分辨率增强的实现类似于莫尔条纹。通过在不同扫描位置用光掩模获取图像，使传统显微镜的光学传递函数带宽增大了一倍[13]。然而，宽场空间结构光照明需要在照明端添加光栅等图案化掩模，不适用于扫描显微镜。时间调制扫描显微镜（SPIN）和空间调制扫描显微镜（SPADE）的提出使得超分辨激光扫描显微镜得以实现。这些方法利用时间和空间调制在点扫描系统中实现了与SIM 相同的效果[14]。SPADE 已经被证明是正确的，也被称为虚拟结构探测(VSD)。由于VSD 需要得到每个扫描点的光斑图像，成像速度受到严重限制[15]。

针对上述缺点，结合线扫描成像方法[16]，将虚拟结构调制应用到大孔径共焦显微镜中（LVSM），以提高共焦显微成像的速度和分辨率。与虚拟结构探测方法的不同之处在于无需后续的移频过程。同时由于共焦显微镜具有独特的切片能力，可以对厚样品成像。不同于其他大部分超分辨成像方法，本方法可以对非荧光样本成像，具有高分辨率、成像速度快的特点。

2 线扫描共焦虚拟结构调制原理

反射式共聚焦相干成像系统如图1 所示，假设照明强度和系统放大倍数为1。在解扫描的情况下，照明光场与样本相互作用后，忽略相位因子，样本表面处的振幅分布为：

式（1）中，hil(x,y)为照明路径的二维振幅点扩散函数，s为样本振幅分布，(x1,y1)为样本所在位置。样本与照明光场作用后的光场振幅分布A1(x,y,x1,y1)经过探测光路后对图像平面上某一点(x2,y2)的振幅贡献值可以表示为：

式(2)中，hde(x,y)为探测路径的二维振幅点扩散函数，A2(x,y,x1,y1,x2,y2)为样本上(x1,y1)处与照明光场作用后产生的振幅分布在图像平面(x2,y2)处的贡献值。所有样本点对该处振幅的贡献值叠加，即图像平面上探测到的振幅值。整个图像平面上振幅分布为：

式(3)中，A3(x1,y1,i,j)为图像平面处的振幅分布。取检测函数D(i,j)：

式(4)中，A(x1,y1)为探测器最终探测到的振幅图像。hil(x,y)与hde(x,y)均为偶函数，应用卷积的定义：

从式(5)可以看出，最终的振幅图像表现为样本不同位置振幅的叠加，为相干成像过程，系统的成像特性取决于振幅点扩散函数：

则系统的相干传递函数(CTF)可以表示为APSF的傅立叶变换：

式(7)中，Hil(fx1,fy1)和Hde(fx1,fy1)为照明和探测端相干传递函数，⊗表示二维卷积，F为傅立叶变换符号。由式(6)知，系统强度点扩散函数(IPSF)可以表示为[17]：

因为系统的成像性能最终受透镜、照明方式以及检测函数限制，下面分别讨论不同情况下系统振幅点扩散函数的形式。

当照明方式为点照明、D(x1,y1)为虚拟针孔时，即点扫描模式。照明和探测路径振幅点扩散函数可以由透镜孔径的傅立叶变换得到，有效探测点扩散函数为探测点扩散函数和孔径函数D(x1,y1)的卷积。

当照明方式为线照明、D(x1,y1)为虚拟狭缝时，即线扫描模式时。相较于点扫描模式，探测路径振幅点扩散函数不变，由于在照明路径上引入了柱面透镜对光斑进行聚焦，照明路径振幅点扩散函数发生了变化，在一个方向上为高斯分布而在另一个方向上为常数分布。假定扫描方向沿X轴方向，即照明点扩散函数在Y方向上是常数分布，在傍轴近似下有[18]：

式中Φ=2πNA/λ，w≪1，所以式(10)中的第一项因子可以忽略。这意味着照明路径的振幅点扩散函数在沿直线方向有一个恒定的激发。

当照明方式为线照明、D(x1,y1)在矩形狭缝的基础上叠加了余弦掩模，即本文提出的LVSM 方法。照明和探测路径振幅点扩散函数与线扫描模式相同，但检测函数D(x1,y1)变为：

式(10)中，p为狭缝宽度，s为CCD 上线斑的长度，在基于上述线扫描点扩散函数模型分析时取无限小。本文主要在线扫描模式的基础上，研究检测函数为式(10)所表示的形式时系统的相干成像特性。由于单个线扫描模式下只能提取当前扫描方向下的超分辨信息，故只考虑一维方向（沿x轴），式(5)可以重写为：

将式(10)代入式(11)，则式(11)的傅立叶变换

式中Uc(fx1)是低频分量、Ul(fx1)和Ur(fx1)是高频分量，这些分量可以很容易的从式(13)中确定。

3 仿真计算

模拟照明激光波长 λ为488 nm，物镜数值孔径为0.4，余弦函数调制频率f0取0.9NA/λ，θ 和φ 均取0。扫描模式为沿X方向，即0°方向。取极限情况，即狭缝宽度s无限小时，按照式(8)对普通线扫描共聚焦的IPSF 仿真，结果如图2（彩图见期刊电子版）所示。根据图2(c)，X方向上的半高全宽(FWHM)为Y方向上的0.71，这是由于在X方向上是共聚焦成像，Y方向上是宽场成像的原因。从图2(b)中的光学传递函数也可以看出，X方向上的截止频率高于Y方向上的截止频率，这说明传递高频信息能力增强。

按照式(7)计算仿真系统CTF，如图3（彩图见期刊电子版）所示，狭缝宽度取一个艾里斑大小。由图3(c)可以看到，在图像扫描方向上，相较于宽场成像(WM)，CLSM 和LVSM 显微镜的截止频率相等，是WM 的2 倍，对应空间域中分辨率提高。在相同的截止频率下，LVSM 中高频信息的比例明显高于CLSM。说明LVSM 具有更强的高频信息传递能力。这归因于余弦函数的调制作用，它提高了高频比。

基于线扫描共聚焦的结构探测方式中，结构检测函数D(x,y)直接作用于探测器平面的解扫描图像，因此无需与样本共轭，检测函数图像保持不变。这与虚拟结构探测(VSD)超分辨方法的不同之处在于，VSD 方法中，结构检测函数作用于非解扫描图像，而成像系统本身是解扫描的，所以结构函数图像需要与样本共轭，以达到非解扫描调制的效果，进而得到扫描成像下类似于宽场SIM 的图像数据。采用宽场SIM 完全相同的重建算法将高频信息移动到正确的位置上，以达到超分辨的效果，而上述方法在线扫描共聚焦的解扫描图像下直接使用检测函数进行调制，系统CTF 的截止频率得到扩展，无需进行移频，即使不采用后续的图像处理过程，图像分辨率仍然有所提高。只是存在图3(c)红色曲线所示的高频信息透过率低的问题，所以提高的效果并不明显。图像重建流程图如图4 所示，可以通过改变调制函数相位建立如式(13)所示的等式，从而解算出沿着这些方向上的频率分量，采用维纳滤波对其进行降噪处理，随后使用广义维纳滤波器对处理后的频域信息进行加权叠加恢复，从而达到增强高频分量的效果。使用广义维纳滤波器进行加权叠加时，各分量权重因子主要取决于不同频率分量的信噪比，可以通过文献[19]中的方法进行估计。将加权叠加后的频域图像傅立叶逆变换到空间域后，得到最终重建的振幅图像，再进行平方运算即得到强度图像。

由于线扫描共聚焦只在一个方向上是共焦成像，利用特定的检测函数只能扩大系统单一方向上CTF 的截止频率。为了验证各向同性分辨率提高的可行性，需要旋转图像场获得不同方向下的线扫描数据，进行虚拟调制。调制方向越多，各向同性分辨率提高越明显，同时成像速率有所降低，但是相较于点扫描方式仍然有很大提升。可以根据实际情况选择合适的调制方向。用结构化检测函数调制每个扫描位置得到线斑图像，计算线斑内图像沿一个方向上的积分，得到一系列代表当前扫描位置的值，这些值和它们对应的位置即代表结构调制后的图像，并用于之后的重建过程。由于调制方式是虚拟调制，数字掩模的方向和相位都可以精确知道，因此不存在调制模式和相位的估计误差。

为了说明上述方法的有效性，本文模拟了LVSM 的成像结果，如图5（彩图见期刊电子版）所示。样本扫描方向取0°和90°。从图5(e)和5(f)可以看出，高频部分信息被截止，样品的细节信息丢失，经过LVSM 重建后，对应方向上的高频信息被移入系统频域通带内并处于正确的位置上，扫描方向上的频谱扩大。将频域图像变换到空间域，由图5(b)和5(c)对比可以看到，分辨率有明显提高。

4 实验与结果

4.1 实验系统

图6 展示了基于激光线扫描结构探测共聚焦显微镜的(LVSM)示意图。采用波长为633 nm单模氦氖激光器产生偏振激光，光衰减器用于降低光的强度，柱面透镜CL（f=180 mm）仅在一个方向上有聚焦特性，将衰减后的激光聚焦成一条直线入射到扫描振镜上，单轴扫描振镜（Mode 6215 CTI）在扫描方向上振动，用于引导聚焦直线通过自制的扫描透镜、筒镜（TTL 180-A Olympus）、物镜，在样本上沿着确定的方向移动。为了减小渐晕效果，控制振镜中心与物镜瞳孔平面共轭。来自样本的反射光被一维扫描系统解扫描，并通过中间光学系统中继到图像平面。探测器收集当前扫描位置的线斑图像，随着扫描振镜的摆动，探测器平面上的解扫描图像位置不变但图像信息不断更新，对应当前扫描位置，连续采集512 条不同扫描位置的线斑图像，得到二维图像。

使用二维图像采集器件sCMOS 采集单个线斑图像（ORCA-Flash4.0 V2，Hamamatsu），相比较EMCCD，sCMOS 具有更高的量子效率和低噪声输出，线斑图像ROI 区域设为512 pixel×64 pixel 时，单个方向理论图像的采集速度可以达到3 206帧/s。实际实验过程中考虑到器件的响应时间和程序的延时，图像采集速率降低，单个方向图像采集时间约为0.25 s。虚拟狭缝取一个艾里斑直径大小，用于对采集到的图像做检测处理。

使用数值孔径为0.13 的4×平场半复消色差物镜（Olympus）。实验中的样本为标准光学分辨率靶（USAF 19511×，Edmund）。为了验证最终的各项同性分辨率提升效果，采用步进电机驱动的电动旋转位移台转动样本，最大转动速率为50°/s，取0°、90°两个图像采集方向，整个图像旋转过程约需要2 s。

4.2 实验结果

图7（彩图见期刊电子版）为采集到的部分线斑图像和最终的图像重建结果。图中蓝色线段标记区域是分辨率测试板Y方向线对的8.2 至8.6 组，黄色线段标记区域为X方向线对的8.5组，绿色线段标记区域为8.6 组。图8（彩图见期刊电子版）为常规线扫描共聚集和线扫描虚拟结构调制共聚焦在特定区域的归一化强度曲线。由图8(a)可知，Y方向上，常规线扫描共聚焦显微镜能分辨8.2 组，不能分辨8.3 组，8.2 组的线对数为287 lp/mm，对应空间周期为3.48μm。LVSM显微镜可以分辨到8.5 组，不能分辨8.6 组，8.5 组的线对数为406 lp/mm，对应空间周期为2.46μm，LVSM 显微镜的分辨率比具有相同狭缝大小的线扫描共聚焦显微镜高。从图8(b)、8(c)可知，X方向上，线扫描共聚焦均不能分辨8.5 组和8.6 组，相比之下，LVSM 显微镜可以分辨到8.5 组，不能分辨8.6 组。因此可知，LVSM显微镜将X方向和Y方向分辨率均提高到2.46μm，是普通线扫描共聚焦显微镜的1.4 倍。

上述实验证明了LVSM 显微镜可以突破衍射极限，同时可以进行高速成像。该理论表明，LVSM 显微镜可以通过2 个方向的调制因子来提高横向分辨率和成像速度。标准分辨率靶的实验证实，基于高速成像的线扫描方式，LVSM 显微镜可以显示常规共聚焦显微镜无法检测到的详细结构。

5 结论

本文提出了一种基于结构调制的线扫描共聚焦显微成像方法,推导了相关理论及重建方法，并进行了实验验证。仿真和实验结果表明，系统CTF 扩大，成像分辨率是普通共焦显微镜的1.4倍。该方法与点扫描光斑虚拟结构调制成像相比，可以大幅度提高系统的成像速率，其只需要2.5 s 即完成两个方向的图像扫描，图像大小为512 pixel×512 pixel。在同样的图像场大小下，后者约需要260 s 来完成数据采集。图像采集速度提高了104 倍。