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圖1 反射式共聚焦系統(tǒng)示意圖

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圖2 IPSF 理論仿真結(jié)果。(a)普通線掃描共聚焦顯微鏡的IPSF，(b)IPSF 的傅立葉變換，(c)圖(a)中X 和Y 方向上的歸一化強度分布

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)圖3 系統(tǒng)CTF 仿真。(a)傳統(tǒng)線掃描共聚焦(CLSM)的CTF，(b)線掃描結(jié)構(gòu)調(diào)制共聚焦(LVSM)的CTF，(c)黑色曲線為(a)中Y 方向歸一化頻率分布(WM)，藍色和紅色曲線分別為(a)、(b)中X 方向歸一化頻率分布(CLSM、LVSM)

Fig.4 Flow chart of image reconstruction圖4 圖像重建過程流程圖

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圖5 線掃描共聚焦虛擬結(jié)構(gòu)調(diào)制仿真（LVSM）。(a)仿真使用的輻條狀樣品。(b)普通共聚焦圖像。(c)取0°、90°兩個掃描方向，結(jié)合對應(yīng)方向上的結(jié)構(gòu)檢測函數(shù)重建后圖像。(d)、(e)、(f)分別是(a)、(b)、(c)的傅立葉變換，即對應(yīng)的頻域圖像

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圖6 實驗系統(tǒng)示意圖

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圖7 分辨率測試目標的線掃描虛擬結(jié)構(gòu)調(diào)制共聚焦實現(xiàn)。(a)掃描方向為0°時，采集的第20、205、360、490 條線斑圖像。(b)掃描方向為90°時，采集的第20、205、360、490 條線斑圖像。(c)掃描方向為90°時，常規(guī)線掃描共聚焦獲得的分辨率測試靶圖片。(d)LVSM 超分辨重建后圖像

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)圖8 常規(guī)線掃描共聚焦和線掃描虛擬結(jié)構(gòu)調(diào)制共聚焦在特定區(qū)域的歸一化強度曲線。圖7(c)、7(d)中(a)藍色線段標記區(qū)域(b)黃色線段標記區(qū)域及(c)綠色線段標記區(qū)域的歸一化強度分布對比

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.

——中文對照版——

1 引言

由于光波衍射效應(yīng)的存在，傳統(tǒng)光學顯微鏡的分辨率受到限制[1-2]。激光掃描共焦顯微鏡（LSCM）具有比寬場顯微鏡更高的分辨率。因為它使用緊密聚焦的激發(fā)光束和針孔檢測來抑制離焦背景光[3-4]。但是受到針孔大小的限制，在共焦顯微鏡中，減小針孔的尺寸在提高分辨率的同時也降低了信噪比。為了保持二者的平衡，針孔尺寸一般較大，這會導致橫向分辨率低于理想結(jié)果，仍處于衍射極限之內(nèi)[5-6]。近20 年來，許多超分辨光學顯微方法得到了推廣，如受激發(fā)射損耗顯微鏡（STED）[7]、結(jié)構(gòu)光照明顯微鏡（SIM）[8]等，這些方法主要基于兩個原理，分別是壓縮點擴散函數(shù)（PSF）和增加光傳遞函數(shù)（OTF）帶寬[9]。除此之外，隨機光學重建顯微鏡（STORM）[10]和光激活定位顯微鏡（PALM）[11]通過使用光開關(guān)熒光探針定位單個分子來實現(xiàn)超分辨率。上述這些方法在熒光標記成像分辨率上都有突破，但也都有其局限性。PALM 和STORM長期以來一直受到成像速度的限制，STED 顯微鏡要求激發(fā)光譜和發(fā)射光譜必須與給定的激發(fā)和耗盡波長相匹配，SIM 只能成像光學薄樣品[12]。

結(jié)構(gòu)探測顯微鏡基于結(jié)構(gòu)照明原理，分辨率增強的實現(xiàn)類似于莫爾條紋。通過在不同掃描位置用光掩模獲取圖像，使傳統(tǒng)顯微鏡的光學傳遞函數(shù)帶寬增大了一倍[13]。然而，寬場空間結(jié)構(gòu)光照明需要在照明端添加光柵等圖案化掩模，不適用于掃描顯微鏡。時間調(diào)制掃描顯微鏡（SPIN）和空間調(diào)制掃描顯微鏡（SPADE）的提出使得超分辨激光掃描顯微鏡得以實現(xiàn)。這些方法利用時間和空間調(diào)制在點掃描系統(tǒng)中實現(xiàn)了與SIM 相同的效果[14]。SPADE 已經(jīng)被證明是正確的，也被稱為虛擬結(jié)構(gòu)探測(VSD)。由于VSD 需要得到每個掃描點的光斑圖像，成像速度受到嚴重限制[15]。

針對上述缺點，結(jié)合線掃描成像方法[16]，將虛擬結(jié)構(gòu)調(diào)制應(yīng)用到大孔徑共焦顯微鏡中（LVSM），以提高共焦顯微成像的速度和分辨率。與虛擬結(jié)構(gòu)探測方法的不同之處在于無需后續(xù)的移頻過程。同時由于共焦顯微鏡具有獨特的切片能力，可以對厚樣品成像。不同于其他大部分超分辨成像方法，本方法可以對非熒光樣本成像，具有高分辨率、成像速度快的特點。

2 線掃描共焦虛擬結(jié)構(gòu)調(diào)制原理

反射式共聚焦相干成像系統(tǒng)如圖1 所示，假設(shè)照明強度和系統(tǒng)放大倍數(shù)為1。在解掃描的情況下，照明光場與樣本相互作用后，忽略相位因子，樣本表面處的振幅分布為：

式（1）中，hil(x,y)為照明路徑的二維振幅點擴散函數(shù)，s為樣本振幅分布，(x1,y1)為樣本所在位置。樣本與照明光場作用后的光場振幅分布A1(x,y,x1,y1)經(jīng)過探測光路后對圖像平面上某一點(x2,y2)的振幅貢獻值可以表示為：

式(2)中，hde(x,y)為探測路徑的二維振幅點擴散函數(shù)，A2(x,y,x1,y1,x2,y2)為樣本上(x1,y1)處與照明光場作用后產(chǎn)生的振幅分布在圖像平面(x2,y2)處的貢獻值。所有樣本點對該處振幅的貢獻值疊加，即圖像平面上探測到的振幅值。整個圖像平面上振幅分布為：

式(3)中，A3(x1,y1,i,j)為圖像平面處的振幅分布。取檢測函數(shù)D(i,j)：

式(4)中，A(x1,y1)為探測器最終探測到的振幅圖像。hil(x,y)與hde(x,y)均為偶函數(shù)，應(yīng)用卷積的定義：

從式(5)可以看出，最終的振幅圖像表現(xiàn)為樣本不同位置振幅的疊加，為相干成像過程，系統(tǒng)的成像特性取決于振幅點擴散函數(shù)：

則系統(tǒng)的相干傳遞函數(shù)(CTF)可以表示為APSF的傅立葉變換：

式(7)中，Hil(fx1,fy1)和Hde(fx1,fy1)為照明和探測端相干傳遞函數(shù)，?表示二維卷積，F(xiàn)為傅立葉變換符號。由式(6)知，系統(tǒng)強度點擴散函數(shù)(IPSF)可以表示為[17]：

因為系統(tǒng)的成像性能最終受透鏡、照明方式以及檢測函數(shù)限制，下面分別討論不同情況下系統(tǒng)振幅點擴散函數(shù)的形式。

當照明方式為點照明、D(x1,y1)為虛擬針孔時，即點掃描模式。照明和探測路徑振幅點擴散函數(shù)可以由透鏡孔徑的傅立葉變換得到，有效探測點擴散函數(shù)為探測點擴散函數(shù)和孔徑函數(shù)D(x1,y1)的卷積。

當照明方式為線照明、D(x1,y1)為虛擬狹縫時，即線掃描模式時。相較于點掃描模式，探測路徑振幅點擴散函數(shù)不變，由于在照明路徑上引入了柱面透鏡對光斑進行聚焦，照明路徑振幅點擴散函數(shù)發(fā)生了變化，在一個方向上為高斯分布而在另一個方向上為常數(shù)分布。假定掃描方向沿X軸方向，即照明點擴散函數(shù)在Y方向上是常數(shù)分布，在傍軸近似下有[18]：

式中Φ=2πNA/λ，w?1，所以式(10)中的第一項因子可以忽略。這意味著照明路徑的振幅點擴散函數(shù)在沿直線方向有一個恒定的激發(fā)。

當照明方式為線照明、D(x1,y1)在矩形狹縫的基礎(chǔ)上疊加了余弦掩模，即本文提出的LVSM 方法。照明和探測路徑振幅點擴散函數(shù)與線掃描模式相同，但檢測函數(shù)D(x1,y1)變?yōu)椋?/p>

式(10)中，p為狹縫寬度，s為CCD 上線斑的長度，在基于上述線掃描點擴散函數(shù)模型分析時取無限小。本文主要在線掃描模式的基礎(chǔ)上，研究檢測函數(shù)為式(10)所表示的形式時系統(tǒng)的相干成像特性。由于單個線掃描模式下只能提取當前掃描方向下的超分辨信息，故只考慮一維方向（沿x軸），式(5)可以重寫為：

將式(10)代入式(11)，則式(11)的傅立葉變換

式中Uc(fx1)是低頻分量、Ul(fx1)和Ur(fx1)是高頻分量，這些分量可以很容易的從式(13)中確定。

3 仿真計算

模擬照明激光波長 λ為488 nm，物鏡數(shù)值孔徑為0.4，余弦函數(shù)調(diào)制頻率f0取0.9NA/λ，θ 和φ 均取0。掃描模式為沿X方向，即0°方向。取極限情況，即狹縫寬度s無限小時，按照式(8)對普通線掃描共聚焦的IPSF 仿真，結(jié)果如圖2（彩圖見期刊電子版）所示。根據(jù)圖2(c)，X方向上的半高全寬(FWHM)為Y方向上的0.71，這是由于在X方向上是共聚焦成像，Y方向上是寬場成像的原因。從圖2(b)中的光學傳遞函數(shù)也可以看出，X方向上的截止頻率高于Y方向上的截止頻率，這說明傳遞高頻信息能力增強。

按照式(7)計算仿真系統(tǒng)CTF，如圖3（彩圖見期刊電子版）所示，狹縫寬度取一個艾里斑大小。由圖3(c)可以看到，在圖像掃描方向上，相較于寬場成像(WM)，CLSM 和LVSM 顯微鏡的截止頻率相等，是WM 的2 倍，對應(yīng)空間域中分辨率提高。在相同的截止頻率下，LVSM 中高頻信息的比例明顯高于CLSM。說明LVSM 具有更強的高頻信息傳遞能力。這歸因于余弦函數(shù)的調(diào)制作用，它提高了高頻比。

基于線掃描共聚焦的結(jié)構(gòu)探測方式中，結(jié)構(gòu)檢測函數(shù)D(x,y)直接作用于探測器平面的解掃描圖像，因此無需與樣本共軛，檢測函數(shù)圖像保持不變。這與虛擬結(jié)構(gòu)探測(VSD)超分辨方法的不同之處在于，VSD 方法中，結(jié)構(gòu)檢測函數(shù)作用于非解掃描圖像，而成像系統(tǒng)本身是解掃描的，所以結(jié)構(gòu)函數(shù)圖像需要與樣本共軛，以達到非解掃描調(diào)制的效果，進而得到掃描成像下類似于寬場SIM 的圖像數(shù)據(jù)。采用寬場SIM 完全相同的重建算法將高頻信息移動到正確的位置上，以達到超分辨的效果，而上述方法在線掃描共聚焦的解掃描圖像下直接使用檢測函數(shù)進行調(diào)制，系統(tǒng)CTF 的截止頻率得到擴展，無需進行移頻，即使不采用后續(xù)的圖像處理過程，圖像分辨率仍然有所提高。只是存在圖3(c)紅色曲線所示的高頻信息透過率低的問題，所以提高的效果并不明顯。圖像重建流程圖如圖4 所示，可以通過改變調(diào)制函數(shù)相位建立如式(13)所示的等式，從而解算出沿著這些方向上的頻率分量，采用維納濾波對其進行降噪處理，隨后使用廣義維納濾波器對處理后的頻域信息進行加權(quán)疊加恢復，從而達到增強高頻分量的效果。使用廣義維納濾波器進行加權(quán)疊加時，各分量權(quán)重因子主要取決于不同頻率分量的信噪比，可以通過文獻[19]中的方法進行估計。將加權(quán)疊加后的頻域圖像傅立葉逆變換到空間域后，得到最終重建的振幅圖像，再進行平方運算即得到強度圖像。

由于線掃描共聚焦只在一個方向上是共焦成像，利用特定的檢測函數(shù)只能擴大系統(tǒng)單一方向上CTF 的截止頻率。為了驗證各向同性分辨率提高的可行性，需要旋轉(zhuǎn)圖像場獲得不同方向下的線掃描數(shù)據(jù)，進行虛擬調(diào)制。調(diào)制方向越多，各向同性分辨率提高越明顯，同時成像速率有所降低，但是相較于點掃描方式仍然有很大提升。可以根據(jù)實際情況選擇合適的調(diào)制方向。用結(jié)構(gòu)化檢測函數(shù)調(diào)制每個掃描位置得到線斑圖像，計算線斑內(nèi)圖像沿一個方向上的積分，得到一系列代表當前掃描位置的值，這些值和它們對應(yīng)的位置即代表結(jié)構(gòu)調(diào)制后的圖像，并用于之后的重建過程。由于調(diào)制方式是虛擬調(diào)制，數(shù)字掩模的方向和相位都可以精確知道，因此不存在調(diào)制模式和相位的估計誤差。

為了說明上述方法的有效性，本文模擬了LVSM 的成像結(jié)果，如圖5（彩圖見期刊電子版）所示。樣本掃描方向取0°和90°。從圖5(e)和5(f)可以看出，高頻部分信息被截止，樣品的細節(jié)信息丟失，經(jīng)過LVSM 重建后，對應(yīng)方向上的高頻信息被移入系統(tǒng)頻域通帶內(nèi)并處于正確的位置上，掃描方向上的頻譜擴大。將頻域圖像變換到空間域，由圖5(b)和5(c)對比可以看到，分辨率有明顯提高。

4 實驗與結(jié)果

4.1 實驗系統(tǒng)

圖6 展示了基于激光線掃描結(jié)構(gòu)探測共聚焦顯微鏡的(LVSM)示意圖。采用波長為633 nm單模氦氖激光器產(chǎn)生偏振激光，光衰減器用于降低光的強度，柱面透鏡CL（f=180 mm）僅在一個方向上有聚焦特性，將衰減后的激光聚焦成一條直線入射到掃描振鏡上，單軸掃描振鏡（Mode 6215 CTI）在掃描方向上振動，用于引導聚焦直線通過自制的掃描透鏡、筒鏡（TTL 180-A Olympus）、物鏡，在樣本上沿著確定的方向移動。為了減小漸暈效果，控制振鏡中心與物鏡瞳孔平面共軛。來自樣本的反射光被一維掃描系統(tǒng)解掃描，并通過中間光學系統(tǒng)中繼到圖像平面。探測器收集當前掃描位置的線斑圖像，隨著掃描振鏡的擺動，探測器平面上的解掃描圖像位置不變但圖像信息不斷更新，對應(yīng)當前掃描位置，連續(xù)采集512 條不同掃描位置的線斑圖像，得到二維圖像。

使用二維圖像采集器件sCMOS 采集單個線斑圖像（ORCA-Flash4.0 V2，Hamamatsu），相比較EMCCD，sCMOS 具有更高的量子效率和低噪聲輸出，線斑圖像ROI 區(qū)域設(shè)為512 pixel×64 pixel 時，單個方向理論圖像的采集速度可以達到3 206幀/s。實際實驗過程中考慮到器件的響應(yīng)時間和程序的延時，圖像采集速率降低，單個方向圖像采集時間約為0.25 s。虛擬狹縫取一個艾里斑直徑大小，用于對采集到的圖像做檢測處理。

使用數(shù)值孔徑為0.13 的4×平場半復消色差物鏡（Olympus）。實驗中的樣本為標準光學分辨率靶（USAF 19511×，Edmund）。為了驗證最終的各項同性分辨率提升效果，采用步進電機驅(qū)動的電動旋轉(zhuǎn)位移臺轉(zhuǎn)動樣本，最大轉(zhuǎn)動速率為50°/s，取0°、90°兩個圖像采集方向，整個圖像旋轉(zhuǎn)過程約需要2 s。

4.2 實驗結(jié)果

圖7（彩圖見期刊電子版）為采集到的部分線斑圖像和最終的圖像重建結(jié)果。圖中藍色線段標記區(qū)域是分辨率測試板Y方向線對的8.2 至8.6 組，黃色線段標記區(qū)域為X方向線對的8.5組，綠色線段標記區(qū)域為8.6 組。圖8（彩圖見期刊電子版）為常規(guī)線掃描共聚集和線掃描虛擬結(jié)構(gòu)調(diào)制共聚焦在特定區(qū)域的歸一化強度曲線。由圖8(a)可知，Y方向上，常規(guī)線掃描共聚焦顯微鏡能分辨8.2 組，不能分辨8.3 組，8.2 組的線對數(shù)為287 lp/mm，對應(yīng)空間周期為3.48μm。LVSM顯微鏡可以分辨到8.5 組，不能分辨8.6 組，8.5 組的線對數(shù)為406 lp/mm，對應(yīng)空間周期為2.46μm，LVSM 顯微鏡的分辨率比具有相同狹縫大小的線掃描共聚焦顯微鏡高。從圖8(b)、8(c)可知，X方向上，線掃描共聚焦均不能分辨8.5 組和8.6 組，相比之下，LVSM 顯微鏡可以分辨到8.5 組，不能分辨8.6 組。因此可知，LVSM顯微鏡將X方向和Y方向分辨率均提高到2.46μm，是普通線掃描共聚焦顯微鏡的1.4 倍。

上述實驗證明了LVSM 顯微鏡可以突破衍射極限，同時可以進行高速成像。該理論表明，LVSM 顯微鏡可以通過2 個方向的調(diào)制因子來提高橫向分辨率和成像速度。標準分辨率靶的實驗證實，基于高速成像的線掃描方式，LVSM 顯微鏡可以顯示常規(guī)共聚焦顯微鏡無法檢測到的詳細結(jié)構(gòu)。

5 結(jié)論

本文提出了一種基于結(jié)構(gòu)調(diào)制的線掃描共聚焦顯微成像方法,推導了相關(guān)理論及重建方法，并進行了實驗驗證。仿真和實驗結(jié)果表明，系統(tǒng)CTF 擴大，成像分辨率是普通共焦顯微鏡的1.4倍。該方法與點掃描光斑虛擬結(jié)構(gòu)調(diào)制成像相比，可以大幅度提高系統(tǒng)的成像速率，其只需要2.5 s 即完成兩個方向的圖像掃描，圖像大小為512 pixel×512 pixel。在同樣的圖像場大小下，后者約需要260 s 來完成數(shù)據(jù)采集。圖像采集速度提高了104 倍。