## Abstract

Single-molecule, localization-based, wide-field nanoscopy often suffers from low time resolution because the localization of a single molecule with high precision requires a low emitter density of fluorophores. In addition, to reconstruct a super-resolution image, hundreds or thousands of image frames are required, even when advanced algorithms, such as compressive sensing and deep learning, are applied. These factors limit the application of these nanoscopy techniques for living cell imaging. In this study, we developed a single-frame, wide-field nanoscopy system based on ghost imaging via sparsity constraints (GISC), in which a spatial random phase modulator is applied in a wide-field microscope to achieve random measurement of fluorescence signals. This method can effectively use the sparsity of fluorescence emitters to enhance the imaging resolution to 80 nm by reconstructing one raw image using compressive sensing. We achieved an ultrahigh emitter density of ${143}\;\unicode{x00B5} {{\rm m}^{ - 2}}$ while maintaining the precision of single-molecule localization below 25 nm. We show that by employing a high-density of photo-switchable fluorophores, GISC nanoscopy can reduce the number of sampling frames by one order of magnitude compared to previous super-resolution imaging methods based on single-molecule localization. GISC nanoscopy may therefore improve the time resolution of super-resolution imaging for the study of living cells and microscopic dynamic processes.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. INTRODUCTION

Fluorescence microscopes are essential tools for the study of biological processes and biological phenomena. However, features with sizes smaller than half of the light wavelength (200–300 nm) cannot be resolved in conventional far-field microscopy because of the optical diffraction barrier. In recent decades, several techniques have been proposed to break the diffraction barrier [1]. These techniques can be divided into two main categories. The first involves techniques that spatially modulate the target with patterned illumination, such as stimulated emission depletion microscopy (STED) [2], reversible saturable optical fluorescence transitions (RESOLFT) [3], and structured illumination microscopy (SIM) [4]. These techniques require point scanning or complex illumination patterns to construct one high-resolution image, which limits the speed of live cell imaging. The second involves techniques that use stochastic single-molecule switching and fluctuation of fluorophores, such as photo-activated localization microscopy (PALM) [5], stochastic optical reconstruction microscopy (STORM) [6], point accumulation for imaging in nanoscale topography (DNA-PAINT) [7], and super-resolution optical fluctuation imaging (SOFI) [8]. These techniques must collect a large number of raw images (thousands to tens of thousands, by wide-field imaging rather than sequential point scanning) to trace fluctuation/blinking information of fluorophores, and ensure that no close-by emitters can be switched on simultaneously, which largely restrict the density of emissive fluorophores in each frame [9]. Therefore, their applicability for live-cell imaging is significantly limited by the low temporal resolution owing to the long sampling time and low emitter density.

To increase the temporal resolution in live-cell imaging, several super-resolution methods have been developed. As scanning-based techniques, STED and RESOLFT achieve a very high temporal resolution by reducing the field of view [10]. As wide-field techniques, live SIM uses nine frames with 3.7 to 11 Hz frame rates to construct one high-resolution output image at 100 nm resolution [11]. The temporal resolution of STORM can be improved either by shortening the sampling time using fast-switching dyes [12] and fast scientific-CMOS cameras [13], or by increasing the emitter density [e.g., by fitting clusters of overlapped spots of approximately ${3{-}4}\;\unicode{x00B5} {{\rm m}^{ - 2}}$ (DAOSTORM)] [14], and using compressed sensing (${8.8}\;\unicode{x00B5} {{\rm m}^{ - 2}}$, CS-STORM) [15], or deep learning (${6}\;\unicode{x00B5} {{\rm m}^{ - 2}}$, Deep-STORM) [16]. Other methods used to process a high density of overlapping emitters are the SPIDER (${15}\;\unicode{x00B5} {{\rm m}^{ - 2}}$) [17], SPARCOM (${35}\;\unicode{x00B5} {{\rm m}^{ - 2}}$) [18], and dictionary learning [19], which use different sparsity representations in the transformation domain for the original signal to obtain sparse information. These methods can reduce the acquisition requirements by between hundreds and thousands of frames; however, they worsen the spatial resolution (30–60 nm) and rely on complex data processing. There is still a strong requirement for a live cell imaging technique that can combine spatial super-resolution with high-speed imaging in large fields of view.

In this study, we develop a wide-field nanoscopy based on ghost imaging via sparsity constraints (GISC) with an 80 nm resolution for single-frame imaging. The technique relies on the naturally sparse fluorescence signals and combines the ghost imaging (GI) method with compressive sensing (CS) theory. CS is a signal processing method that uses the sparsity property of the signal to achieve super-resolution imaging with random measurements [20–22]. GI was first demonstrated by the correlation of entangled two-photon states in 1995 and then by the intensity fluctuation correlation of thermal light in 2004 [23–26]. Thermal light GI [23,25,27,28], which was developed in the past decades, can be classified as illumination with pseudo-thermal light and true thermal light. Fluorescence is a true thermal light. GI with true thermal light has been demonstrated by detecting the temporal fluctuation of light fields, and the imaging information was obtained from the second-order correlation of the intensity of light fields at the reference arm and the test arm in the time domain [27,28]. However, such GI requires multiple measurements in the time domain with a single-pixel (bucket) photodetector, which makes single-shot imaging impossible. A GISC camera [29] uses a spatial random phase modulator to modulate the true thermal light from a point position into a spatial fluctuating pseudo-thermal light field and uses a high spatial-resolution detector to detect the intensity information of an object in the spatial domain instead of the detection by the bucket detector in the time domain. The images of the object are obtained by the second-order spatial mutual correlation of light fields between the spatial intensity fluctuations in the predetermined reference arm and the test arm in a single-shot measurement. Moreover, the GI method to extract the imaging information is based on “global random” measurements, and the fluctuating light field follows a Gaussian statistical distribution, which satisfies the restricted isometry property (RIP) required for CS [20,21]. Therefore, as a combination of GI and CS, which is a combination of optical imaging with modern information theory, GISC has led to many potential applications, including super-resolution imaging [30–32], GISC spectral cameras [29], GISC lidars [33], and x-ray diffraction imaging [34].

GISC nanoscopy applies a spatial random phase modulator in a conventional fluorescence microscope to form random speckle patterns and code the detected fluorescence signals. A random measurement matrix composed of speckles corresponding to different positions of the sample plane is built, which transforms the measurement matrix of the signal to satisfy the CS requirement. Therefore, the technique can effectively reconstruct a super-resolution image via sparsity constraints from one low-resolution wide-field speckle image. Through simulations and experiments, we demonstrate the reconstructions with considerably improved spatial resolution, by reducing the number of required exposures to a single-frame exposure, and speeding up the fluorescence imaging process. The technique can be applied to any fluorescent specimen without complex illumination modes or the intrinsic blinking/fluctuation mechanism of fluorescent molecules. At the same time, we achieve ultrahigh density (up to ${143}\;\unicode{x00B5} {{\rm m}^{ - 2}}$) of single-molecule localization with intensity and position information, and the localization precision is still below 25 nm. Therefore, we can combine the proposed technique with single-molecule, localization-based, super-resolution techniques (GISC-STORM), achieving sampling times that are orders-of-magnitude shorter than those of previous approaches without compromising the spatial resolution. Moreover, the signal-to-noise ratio (SNR) and contrast (C), which are the main factors that influence the quality of super-resolution imaging, are investigated.

## 2. METHODS

#### A. Theory

The fluorescent light field is modulated into a speckle pattern by a random phase modulator mounted in front of the detector. GISC nanoscopy consists of two stages: (1) the calibration process, where we collect the speckle patterns $I_{(i^\prime ,j^\prime )}^r(i,j)$ of each position with a point source in the whole sample plane as the prior information, which can be either measured online or be predetermined; and (2) the imaging process, where we can obtain the speckle pattern intensity distribution $ I_{(i^\prime ,j^\prime )}^t $ of all fluorophores within the imaged specimen to achieve super-resolution image reconstruction by calculating the second-order intensity correlation between the calibrated speckle patterns and the imaging speckle with the specimen. The second-order correlation function is expressed as [26,29]

In GISC nanoscopy, the imaging resolution is determined by three aspects. The first is the resolution of GI, which is related to the mutual correlation function of speckle patterns and can be optimized by adjusting the position of the random phase modulator between the tube lens and detector and the parameters of the random phase modulator [29]. Figure 2(a) presents the mutual correlation function of the speckle and point spread function (PSF) of fluorescence signals measured in the experiment. As shown in the figure, the resolution of GI can be improved by a factor of $\sqrt 2 $ compared to conventional wide-field microscopes, which is the same as the resolution of confocal microscopes. The potential resolution of GI has been demonstrated theoretically based on the second-order correlation of light fields [35,36], and can be further improved by calculating the high-order correlation. The second aspect that determines the resolution of GISC nanoscopy is the sparse constraint, which is used as an image processing method to recover high-frequency information lost in the measurement process. A potential twofold increase in the image resolution based on the sparsity constraint has been demonstrated mathematically by Candès [22,32]. The third aspect is the shift with subpixel precision, which allows super-resolution reconstruction from observed multiple low-resolution measurements [37]. Therefore, the higher resolution of GISC nanoscopy can be achieved by using the difference in speckle centers from its subpixel precision random movement at each sampling.

CS theory has proven that imaging information with sparsity can be accurately reconstructed when two key conditions are satisfied: the imaging object’s sparsity in the representation basis and the random measurement. The “global random” measurement of GI and the fluctuating light field obeying a Gaussian statistical distribution strictly satisfies the RIP required for CS. Therefore, we reformulate Eq. (1) in the CS framework to realize single-frame, super-resolution imaging. The original signal intensity $I(i,j)$ is denoted as a column vector $X( {N \times 1,N = {N_i} \times {N_j}} )$. The speckle intensity $ I_{(i^\prime ,j^\prime )}^t $ detected during the imaging process is arranged as a column vector $Y( {M \times 1,M = {M_{i^\prime }} \times {M_{j^\prime }}} )$. After $N$ calibration measurements, we can build a measurement matrix $A( {M \times N} )$; each of the calibrated speckle intensities $I_{(i^\prime ,j^\prime )}^\prime (i,j)$ is reshaped as one column vector of the matrix $A$. The CS process is expressed mathematically as [20]

where $Y$ is the observation of a set of “global random measurements,” $A$ is the random measurement matrix, and $X$ is the original image. Moreover, combining the random measurement matrix and the sparsity fluorescence signals enables compressive sensing to reconstruct molecules more accurately. According to compressive sensing theory, the original information can be reconstructed accurately if [20,21]We use two common compressive sensing algorithms to solve Eq. (2); namely, the gradient projection for sparse reconstruction (GPSR) to solve convex optimization equations [38] and the orthogonal matching pursuit (OMP), which is a greedy iterative algorithm [39]. When using GPSR, Eq. (2) can be transformed to implement the L1 norm minimization: $\min {\left\| x \right\|_1}{\rm s.t.}{\left\| {Ax - y} \right\|_2} \le \varepsilon $ ($\varepsilon $ is related to noise). Then, it can be simplified to a quadratic programming equation. GPSR has advantages in noise reduction and high reconstruction accuracy. For the OMP algorithm, considering the noise, the relationship between the SNR and reconstruction error is expressed as [40]

#### B. Experimental Setup and Calibration Process

Figure 1 shows the experimental setup and imaging process of GISC nanoscopy. We performed the experiment on a conventional inverted microscope (IX83, Olympus, USA) with a total internal-reflection excitation scheme (cellTIRF-4Line, Olympus) and an oil-immersion objective (Olympus, ${100\times}$, Numerical aperture: 1.49). A spatial random phase modulator (DG10-1500-A, Thorlabs) was mounted after the focal plane of the imaging system to generate random speckle patterns of fluorescence signals, and subsequently, a low magnification objective (UPlanSApo ${10\times}$, Olympus) was used to magnify the speckles, which are detected as the matrix $Y$ using a scientific CMOS (Prime 95B, Photometrics, pixel size of 50 nm in the sample plane). Two lasers with wavelengths of 532 nm (OBIS-532 nm-LS-80 mW, Coherent) and a 640 nm (OBIS-640 nm-LX-75 mW, Coherent) were used as different excitation sources, which were filtered by a bandpass excitation filter (ZET532/640x, Chroma) and reflected by a multiband dichroic mirror (ZT532/640rpc, Chroma) onto the sample. The fluorescence light collected from the sample was filtered by a bandpass emission filter (ZET532/640m-TRF, Chroma). To shorten the calibration and reconstruction time, we use the block calibration and reconstruction method. We divided the sample plane into multiple blocks, and analyzed and optimized these blocks separately; the size of each block was ${1.72}\;{ \unicode{x00B5}{\rm m}}\; \times \;{1.72}\;{ \unicode{x00B5}{\rm m}}$. Each block was subdivided into ${86}\; \times \;{86}$ grids as the matrix $X$ (20 nm per grid) to ensure imaging accuracy (According to Nyquist criterion). We took one block as an example in the simulations and experiments.

A single microsphere (20 nm) in the calibration area is selected as the calibration source, which is mounted on a nanopositioning stage (E725, PI). A series of speckle images of microspheres at different positions (spacing of 100 nm) are obtained accurately by controlling the stage and sCMOS synchronously. Then, the normalized measurement matrix $A$ is built with a pixel spacing of 20 nm (which matches the grid size of the matrix $X$) by bilinear interpolation from a series of speckle images. To ensure the accuracy of the reconstruction, two key issues should be considered in the calibration process. The first is the selection of the calibration source, which requires high fluorescence brightness, high stability, and a small difference in the size and spectra with the dye used in the imaging process. Therefore, we use microspheres (20 nm) as calibration sources because their size difference with fluorophores is negligible in terms of the effects on the noise and pixel size of the detector, as demonstrated by a system simulation or deconvolution, and the spectral difference is limited by the bandpass filter selected according to the spectral resolution of the GISC imaging system [29]. Thus, any fluorescent sample can be used at the same excitation wavelength with the microsphere, which makes this technique more widely used in sample preparation and dye selection. The second key issue that should be considered in the calibration process is the translation invariance; in the calibration process, the accuracy of the translation, and the stability of the light source must be guaranteed. Therefore, the mutual correlation between the two adjacent speckle images is set to above 0.95 to reduce the interpolation error. The axial drift must be corrected by a device (IX3-ZDC2, Olympus), and the lateral drift to reconstruct the image is calibrated according to the drift curve of the system obtained before the calibration process.

## 3. RESULTS

#### A. Simulation with the Real Speckle Patterns

We use the real speckle patterns obtained in the calibration process to evaluate the resolution of this technique and its influencing factors. To quantify the improved resolution of our technique, we generated rings with different spacings (80, 120, and 240 nm) composed of high-density molecules. The number of photons emitted from a molecule follows a log-normal distribution (there is a maximum of 4000 photons with a standard deviation of 1700 photons), which matches the experimentally measured single-molecule photon distribution of Alexa Fluor 647. Then, the photon number is multiplied by the peak attenuation coefficient due to the photon diffusion when passing through the phase modulator. As observed in Fig. 2(b), the single-frame reconstruction results with real speckle patterns show that the spatial resolution of GISC nanoscopy is as high as 80 nm according to the Rayleigh criterion; these results represent a significant improvement compared to the corresponding wide-field imaging.

To clarify the factors that influence the spatial resolution in GISC nanoscopy, the SNR and ${ C}$ were investigated. The SNR is the main factor influencing the compressive sensing algorithm, and can be expressed as $ {\rm SNR} = 10{\log _{10}}( {{{{{\bar I}_0}} \mathord{/ {\vphantom {{{{\bar I}_0}} {{{\bar I}_n}}}} } {{{\bar I}_n}}}} ) $, where ${\bar I_0}$ and ${\bar I_n}$ are the mean values of the speckle intensity and the detected noise intensity, respectively; $C = {\sigma _I}/{\bar I_{0}}$ is the key metric to evaluate the characteristics of speckle [41], where ${\sigma _I}$ is the standard deviation of the speckle intensity. $C$ represents the level of fluctuation of the speckle intensity, and the $C$ of the detected speckle pattern is mainly related to the sparsity $K$ of the fluorescence emitters.

To evaluate the effects of different SNRs and Cs on the spatial resolution, we obtained the minimum separation distance of molecules under varying emitter densities (corresponding to different Cs) and SNRs by adding different levels of background noise to the speckle image. The additive noise follows a Gaussian distribution, which matches the noise experimentally detected by the sCMOS. In Fig. 2(c), the effect of the SNR on the resolution is negligible when $C$ is 0.95, which corresponds to an emitter density of ${5 {-} 8}\;\unicode{x00B5} {{\rm m}^{ - 2}}$. However, the SNR has a significant effect on the resolution (which reaches 180 nm at an SNR of 20) when $C$ is 0.75, corresponding to an emitter density above ${100}\;\unicode{x00B5} {{\rm m}^{ - 2}}$. The threshold for accurate reconstruction is an SNR of 20 and a $C$ of 0.75.

Instead of obtaining the continuous intensity distribution of the original signal by the GPSR algorithm, we use the OMP algorithm to achieve an exact molecule localization expressed in discrete grid points with intensity information; the OMP algorithm could identify almost all completely overlapped emitters shown in Fig. 3(a). The image resolution is determined by both the density of identified molecules (through the Nyquist sampling criteria) and the localization precision of molecules [15]. The resolution of 80 nm corresponds to an emitter density of ${156}\;\unicode{x00B5} {{\rm m}^{ - 2}}$. However, in the actual simulation, the molecule identification efficiency was slightly worse due to the limitations of the OMP algorithm and the SNR. To quantify the ability to increase the emitter density and test the robustness of this technique against noise, we randomly generated molecules with different densities in an ${86}\; \times \;{86}$ pixel area under different SNR values, with the number of photons matching that in the experiment of Alexa-647. For each SNR value, we analyzed the density of identified molecules and the localization precision as the density increased. In Fig. 3(b), the maximum density of recovered molecules is ${143}\;\unicode{x00B5} {{\rm m}^{ - 2}}$ at a high SNR of 60, and relatively worse densities of 76 and ${62}\;\unicode{x00B5} {{\rm m}^{ - 2}}$ were obtained at SNRs of 40 and 20, respectively. The localization precision at high density was still 25 nm (${\rm SNR} ={60}$). Although the lower SNR led to a worse localization precision, at very low SNRs the localization precision did not exceed 60 nm, which indicates that the localization precision does not limit the spatial resolution (80 nm). Therefore, the ultrahigh density of molecules is sufficient for single-frame super-resolution imaging with the clearly distinguished structures and features of the sample, and the high localization precision can maintain the imaging resolution. The performance of GISC nanoscopy while increasing the localization density and precision was orders of magnitudes higher compared to those of other high-density localization methods for super-resolution imaging.

#### B. Experimental Imaging

We demonstrate the performance of GISC nanoscopy on single-frame super-resolution imaging of 270 nm (GATTA-Confocal 270R, Atto 647N, Gattaquant) and 160 nm (GATTA-SIM 160R, Alexa 647, Gattaquant) rulers. Samples were imaged using a 532 or 640 nm laser with an intensity of approximately ${0.1 {-} 0.3}\;{\rm kW}\;{{\rm cm}^{ - 2}}$. A 20 nm microsphere (535/575 nm or 660/680 nm) was used for calibration before the imaging process. Single-frame, super-resolution images were reconstructed by the OMP algorithm from one speckle image of the sample. Figure 4(a) shows the reconstruction results of the image of the 270 (upper row) and 160 (lower row) nm rulers with different densities and a comparison with the corresponding wide-field images. Figure 4(b) shows the histogram and the normalized intensity distribution along the white line in the reconstruction results. The intensity distribution represents the reconstruction error distribution and intensity information. Correspondingly, Fig. 4(c) gives a statistical analysis of the accuracy of 40 reconstructed images of the rulers. For the image of the 270 nm ruler, we achieved a small reconstruction accuracy (${270}\;{\rm nm \pm 10}\;{\rm nm}$), and a slightly lower reconstruction accuracy (${160}\;{\rm nm \pm 30}\;{\rm nm}$) with more artifacts for the image of the 160 nm ruler. The reconstruction error and artifacts for the nanometer ruler were derived from the error of the ruler production and the low SNR (corresponding to ${\rm SNR} = {20}$ and ${C} = {0.75}$).

#### C. Fast GISC-STORM

The ultrahigh density localization can not only achieve single-frame, super-resolution imaging but it also can develop a fast nanoscopy method combined with STORM (GISC-STORM), which uses stochastically blinking emitters of high density to improve the spatial resolution compared to that of single-frame imaging. The performance of GISC-STORM is characterized by the identified density and the localization precision of molecules. We performed simulations to evaluate these two metrics with molecule densities ranging from 1 to ${12.5}\,\,\unicode{x00B5} {{\rm m}^{ - 2}}$ and compared the results to those of other molecule identification methods, such as CS-STORM [15] and the single-molecule fitting method (implemented using ThunderSTORM [42]). As shown in Fig. 5(a), GISC-STORM outperforms the other methods in terms of high-density localization. ThunderSTORM only identifies molecules at a maximum density of ${1}\,\,\unicode{x00B5} {{\rm m}^{ - 2}}$ then shows a decreasing trend with further increasing molecule density; CS-STORM follows a similar trend when its molecular density exceeds ${9}\,\,\unicode{x00B5} {{\rm m}^{ - 2}}$. However, the molecule density identified by GISC-STORM increases linearly up to the maximum density of ${143}\,\, \unicode{x00B5} {{\rm m}^{ - 2}}$ [Fig. 3(b)], which is up to 16 and 135 times larger than that of CS-STORM and ThunderSTORM, respectively. For localization precision, Fig. 5(b) shows that, as the molecule density increases, GISC-STORM maintains a localization precision lower than 20 nm, while the others show continuously increasing trends (up to precisions of 98 and 114 nm for CS-STORM and ThunderSTORM, respectively). Therefore, the high localization density and precision of GISC-STORM imply a large reduction in the required sampling frames for the reconstruction of a super-resolution image, and the localization precision does not limit the spatial resolution at high emitter densities. To further assess the sampling speed of GISC-STORM, we compared it with that of CS-STORM and ThunderSTORM qualitatively by testing the required minimum number of frames to reconstruct a given ring with a spacing of 60 nm. The ring is composed of blinking emitters, and the emitter densities per frame are determined by their respective identified molecule density (shown in the above simulation results). Figure 5(c) shows that GISC-STORM can achieve a resolution of 60 nm from 10 frames. Therefore, GISC-STORM is approximately 50 and 400 times faster than CS-STORM, and ThunderSTORM, respectively. In the experiment, GISC-STORM could resolve the 40 nm ruler (GATTA-PAINT 40R, Atto 655, Gattaquant) from only 100 frames, as shown in Fig. 5(d). Moreover, GISC-STORM takes approximately 1–2 min to reconstruct a super-resolution image from 100 frames. Therefore, GISC-STORM can significantly improve the temporal resolution (only need tens to hundreds of frames) without compromising its spatial resolution, making it suitable for application in living cells imaging.

## 4. CONCLUSION

We developed a single-frame, wide-field nanoscopy based on GISC by a random phase modulator to obtain speckle patterns of fluorescence signals. The proposed method can provide a resolution of 80 nm at a high SNR. The ability to improve the resolution is demonstrated by both simulations and experiments. Moreover, the proposed method can reconstruct super-resolved structures with high emitter densities and low computational cost using the OMP algorithm of CS; the density can reach ${143}\,\,{{\rm \unicode{x00B5}{\rm m}}^{ - 2}}$, and the localization precision remains below 25 nm, which is a significant improvement compared to other super-resolution microscopes based on the localization of a single molecule.

GISC nanoscopy not only enables ultrafast super-resolution imaging down to a single frame, but can also obtain high resolution by combining it with STORM; it significantly increases STORM imaging speed, which can promote the application of super-resolution imaging in the study of living cells and microscopic dynamic processes. Moreover, this technique can be used for any fluorescent sample without variations in brightness induced either by photoactivation or intrinsically. The instruments required for imaging can easily be combined with a conventional microscope.

## Funding

National Key Research and Development Program of China (2016YFC0100603); Science and Technology Commission of Shanghai Municipality (18DZ1100403); National Natural Science Foundation of China (11674337).

## Acknowledgment

We would like to thank Y. Liu (iHuman Institute of ShanghaiTech University) and W. Ji (Institute of Biophysics, Chinese Academy of Sciences) for helping with the sample and reading of the paper, and J. Ma (Fudan University) for discussion and helpful comments.

## Disclosures

The authors declare that there are no conflicts of interest related to this paper.

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