2.1.

Numerical Simulations

The simulations were implemented using custom-built codes in MATLAB 2013a (Mathworks Inc.). A resolution test object featuring the spatial frequencies increase from outer circle to inner circle was generated for subsequent evaluations. The test object consists of 40 lines with different angles. Although SOFI tolerates higher excited molecule density than PALM/STORM, it also fails to reconstruct faithful images at ultrahigh density regime. The central region of the test object was removed owing to its ultrahigh spatial frequency, thus avoiding severe artifacts induced by SOFI processing. The spatial frequency of the test object ranges from $2\mu {\mathrm{m}}^{-1}$ to $8\mu {\mathrm{m}}^{-1}$. The distribution of spatial frequency decreases from the inner circle ($8\mu {\mathrm{m}}^{-1}$) to outer circle ($2\mu {\mathrm{m}}^{-1}$). The image sequence was generated with multiple blinking points distributed on the test object. The blinking statistics obeys the power law distribution, i.e., $P(t)\propto t^{-\alpha }$, where $P$ is the probability density and $\alpha$ denotes the power law exponent. In the simulations, the exponents for on and off blinking are 1.5 and 1.9, respectively. Shot noise, read noise, and fixed pattern noise were also selectively added to the generated image sequence. The SOFI processing was implemented using the bSOFI toolbox developed by Marcel Leutenegger.²⁰ The cross-correlation evaluation was performed using the custom-built MATLAB program. To estimate the fidelity of SOFI reconstruction under different situations, we have calculated the cross-correlation coefficients between the simulated SOFI reconstruction and the test object convolved with corresponding PSFs. We converted the image to the polar coordinate first, and take 360 deg with each radius, corresponding to one spatial frequency. The correlation values for different situations were calculated using MATLAB R2013a (MathWorks). We evaluated the performance of the SOFI reconstructed images based on the calculation of multiple correlation values along the circular directions at multiple radial values of the test object. Finally, the curve showing the correlation value versus spatial frequency was plotted to estimate the SOFI imaging performance.

2.2.

Data Processing

A subpixel drift-correction algorithm based on discrete Fourier transforms and nonlinear optimization in a self-reorganized code written in MATLAB 2013a was used to correct for drift of raw data during image acquisition.³⁴ High-order spatiotemporal crosscumulants between adjacent pixels was calculated during SOFI processing. The nonlinear brightness response in SOFI images can be linearized and the spatial resolution can be further improved linearly with the cumulant order after balancing. In this paper, fourth-order cross-cumulant balanced SOFI (bSOFI) was applied to 1200 frames using the shortest accessible lag time.^20,28 The pixel sizes of average images were interpolated the same as the fourth-order SOFI images.

2.3.

Cell Culture, Transfection, and Fixation

Cos-7 cells were plated on a glass bottomed dish and cultured in Dulbecco’s modified eagle medium supplemented with 10% fetal bovine serum for 2 days. For optimal microtubule imaging, cells were preextracted with extracting buffer containing 0.1 M piperazine-1,4-bisethanesulfonic acid (PIPES), 1 mM ethylene glycol tetraacetic acid (EGTA), $1\mathrm{mM}{\mathrm{MgCl}}_2$, and 0.2% Triton X-100 for 1 min. Cells were then fixed in phosphate-buffered saline (PBS) with 4% paraformaldehyde (PFA) and 0.1% glutaraldehyde (GA) for 15 min at room temperature. After fixation, cells were permeabilized with 0.5% Triton X-100 for 5 min, washed with PBS and quenched for 7 min with 0.1% sodium borohydride (${\mathrm{NaBH}}_4$), then washed again and blocked with 5% bovine serum albumin and 0.1% Triton X-100 for 30 min. Biotin conjugated antialpha tubulin antibody (Abcam#ab74696) was diluted in the blocking buffer and incubated for 1 h at room temperature. Cells were washed three times from the primary antibody with PBS. Streptavidin conjugated Qdots 705 (Invitrogen) were diluted in the same blocking buffer and incubated with cells for 1 h at room temperature. Samples were stored at 4°C before imaging.

2.4.

Imaging System and Acquisition

For comparison of SNR between sCMOS and EMCCD in SOFI imaging, a commercial optical microscopy system (Ti-E; Nikon, Japan) with a high-NA oil objective ($60\times$, NA 1.45 Nikon with internal $1.5\times$ magnification lenses in the microscope system) was used for imaging acquisition, as described in Sec. 4.1. A 473-nm laser (MXL-III-473, CNI laser, China) was used to excite Qdots 525. A band-pass filter (500 to 550 nm) was placed before an EMCCD camera (Andor iXon DU-897) and scientific complementary metal-oxide-semiconductor (sCMOS, Andor Zyla 4.5). The maximum power near the back pupil of the objective was about 10 mW for the 473-nm laser in Fig. 7. The excitation intensity for the 473-nm laser is about $60\mathrm{W}/{\mathrm{cm}}^2$ on specimen. The same commercial optical microscopy system (Ti-E; Nikon, Japan) with a high-NA oil objective ($100\times$, NA 1.49 Nikon with internal $1.5\times$ magnification lenses in the microscope system) was used to compare SOFI performance between sCMOS and EMCCD for microtubule imaging in Sec. 4.2. A 488-nm fiber laser (MPB Communications, 200 mW) was used to excite the microtubule labeled with Qdots 705. A polychromatic mirror set (Di01-R405/488/561/635-Dichroic and FF01-446/523/600/677-Emission) was used before the EMCCD (Andor iXon DU-897) and sCMOS (Hamamatsu ORCA-Flash 4.0). The maximum power near the back pupil of the objective was about 15 mW for the 488-nm laser in Fig. 8. The excitation intensity for the 488-nm laser is about $90\mathrm{W}/{\mathrm{cm}}^2$ on specimen.

3.1.

Comparison of Super-Resolution Optical Fluctuation Imaging Performances on Different Optical Pixel Sizes

In this simulation, we compared the effect of pixel sizes on SOFI imaging. We assumed a fluorescence wavelength of 625 nm and a numerical aperture (NA) of 1.4 with different pixel sizes. SOFI images were reconstructed based on the auto (cross-) correlation or cumulants of the intensity fluctuation. For example, the second-order autocorrelation function $G_2(r,\tau )$ is given by^12,35

The two-dimensional detector was employed to record the fluorescence fluctuation. The final SOFI image result can thus be expressed with the calculation of corresponding pixel-wise cumulants (correlations). For example, the second-order SOFI pixel value $F(x,y)$ can be expressed as

As shown in Fig. 1, multiple blinking emitters distributed on a simulated resolution test object ($\mathrm{width}=10\mu \mathrm{m}$) with different pixel sizes were generated for SOFI reconstruction. Due to the optical diffraction, the averaged image is blurred compared to the test object [Figs. 1(a)–1(c)]. Subsequently, we implemented third-order SOFI processing for super-resolution image reconstruction of the raw data with different pixel sizes. As can be seen, the third-order SOFI image [Figs. 1(e)–1(g)] can significantly enhance the spatial resolution. However, a close-up look at the dense region reveals that the SOFI result obtained with smaller pixel size (65 nm) has better resolving capability than the counterpart obtained with larger pixel size (160 nm) shown in Figs. 1(e)–1(g).¹⁸ The cross-section profiles confirm that smaller pixel size exhibits higher resolution enhancement than larger ones in SOFI imaging [Fig. 1(h)]. We estimated that the spatial resolutions of the third-order SOFI with pixel sizes of 65, 108, and 160 nm are 110, 150, and 155 nm, respectively.

Fig. 1

Simulations and comparison of SOFI reconstructions with different optical pixel sizes. (a)–(c) The averaged conventional images of the test object with pixel sizes of 65, 108, and 160 nm, respectively. Insets: magnified regions. (d) The test object. Dotted circles indicate the positions of different spatial frequencies. (e)–(g) Third-order SOFI images with a pixel size of 65, 108, and 160 nm, respectively. Insets: magnified regions. (h) Cross sections indicated by the white lines in (c)–(g). (i) Correlation value versus spatial frequency for (e)–(g), respectively. (j) Simulation of the curve representing cut-off frequency (spatial resolution) as a function of the pixel size; 500 frames were simulated for the SOFI reconstruction. This simulation is performed with shot noise included and fixed photon budget ($\sim \mathrm{150,000}$ photons for each frame) for different pixel sizes; 500 photons are detected from a single molecule in the bright state per time unit. In our simulations, the SNR is about 3 when 40- to 50-nm pixel sizes were used.

The cross-correlation values between the SOFI reconstructed image and the original image were calculated to estimate the SOFI performance. The correlation values were calculated between the third-order SOFI reconstructed images and the target convolved with the third-order SOFI PSF. As can be seen in Fig. 1(i), the image with the pixel size of 160 nm presents poor correlation with the target. With the reduction of pixel size, the correlation value enhances accordingly. The image with 65-nm pixel size presents a high correlation relationship with the target. The cut-off frequency is defined as the spatial frequency when the correlation coefficient drops to 0.2. The value of cut-off frequency is used for evaluating the performance of SOFI imaging in the simulation. The corresponding cut-off frequencies were estimated to be 3.8, 3.2, and $2.4\mu {\mathrm{m}}^{-1}$ for pixel sizes of 65, 108, and 160 nm, respectively.

We simulated the curve investigating the relationship between the cut-off frequency and pixel sizes. As can be seen in Fig. 1(j), the best SOFI result can be obtained with pixel sizes of $\sim 40$ to 50 nm. When increasing the pixel sizes, the cut-off frequencies drop monotonically. When further reducing the pixel sizes ($<25\mathrm{nm}$), the cut-off frequency also decreases due to lower SNR (less photons being collected) for smaller pixel sizes. Note that the optimal pixel size obtained here is dependent on the SNR of the generated image sequences. In this simulation, the SNR is about 3 when 40- to 50-nm pixel sizes were used.

3.2.

Comparison of Super-Resolution Optical Fluctuation Imaging Performances on Different Frame Rates

The frame rate of a detector also influences SOFI reconstruction, especially in live-cell applications. In general, sCMOS is able to achieve more than $100\text{frame}/\mathrm{s}$ (fps) with a large field-of-view ($2048\times 2048\text{pixels}$), much faster than EMCCD. A filamentous network mimicking the cytoskeleton in a living cell was generated for this simulation [Fig. 2(a)]. In order to simulate the dynamic motion of the cytoskeleton in a living cell, the simulated network structure was continuously rotated for 1 deg counterclockwise per second (rotation occurs every frame). We first compared reconstructed SOFI performances with the same frame number but different frame rates. The frame number used for SOFI reconstruction is 100 frames. The frame rates are 100 and 33 fps, respectively. The reconstructed super-resolution images were obtained by the second-order SOFI processing using 100 raw frames. SOFI images obtained with 100 fps exhibit better continuity and improved spatial resolution compared with the counterpart obtained with 33 fps [Figs. 2(b)–2(e)]. Due to the designed sample movement, the low frame rate images show an obvious filamentous split artifact [Figs. 2(c) and 2(e)], which is caused by the threefold image acquisition time during image collection. This suggests that the slow acquisition speed can lead to a larger artifact from the inevitable sample movement in live-cell SOFI imaging.

Fig. 2

Comparison between SOFI imaging with different frame rates. Hundred frames are used for SOFI reconstruction. The simulation was performed with shot noise included, the collected photon numbers for each frame were dependent on frame rates. (a) Averaged conventional image of a simulated filamentous network. (b) Second-order SOFI image obtained with 100-fps frame rate and 1-s acquisition time. (c) Second-order SOFI image obtained with 33-fps frame rate and 3-s acquisition time. (d) and (e) Magnified regions indicated by the white dotted box in (b) and (c). (f) Cross sections indicated by the white dotted lines in (a), (d), and (e). Scale bars: Upper panel: $2\mu \mathrm{m}$. Lower panel: $1\mu \mathrm{m}$.

In Fig. 3, the comparison of SOFI reconstruction with different frame rates at the same acquisition time was simulated. In this simulation, the acquisition time for different frame rates was always set to 1 s. The rotation of the simulated filamentous network was $1\mathrm{deg}/\mathrm{s}$, with the rotation occuring in each image frame. The frame rates range from 10 fps to 600 fps. As can be seen, the SNRs of raw image frames decline with the increase of frame rates. However, the SOFI reconstruction results for high frame rates present better performance than the results for low frame rates. During the image acquisition time of 1 s, the collected image numbers for low frame rates are insufficient for reliable SOFI reconstruction. For high frame rates, although the SNRs of collected images decrease, the SOFI reconstructed results exhibit better performance with sufficiently collected image numbers.

Fig. 3

Comparison of SOFI imaging with different frame rates at constant acquisition time of 1 s. (a)–(e) Generated single frame images of a simulated filamentous network with frame rates of 10, 33, 100, 200, and 600 fps, respectively. (f)–(j) Second-order SOFI images of the simulated filamentous network with frame rates of 10, 33, 100, 200, and 600 fps, respectively. (k) Cross sections indicated by the white dotted lines in (f)–(j). This simulation is performed with shot noise and read noise included for different frame rates. The collected photons for each frame decrease linearly with the increase of frame rates.

3.3.

Comparison of Super-Resolution Optical Fluctuation Imaging Performances Using sCMOS and EMCCD with the Same Optical Pixel Size and the Same Frame Rate

In this simulation, we evaluated the influences of multiple types of noise on the quality of SOFI images. A comparison between sCMOS and EMCCD with different noise parameters was implemented as shown in Fig. 4. As sCMOS and EMCCD encompass distinct noise levels at different photon numbers, we conducted this simulation based on the parameters from the elaborate experimental studies by Huang et al.^36,37 In the simulation, both pixel-independent (shot noise) and pixel-dependent noises (including fixed pattern noise and read noise) were taken into account.³⁸ Shot noise follows Poisson distribution, while pixel-dependent noise commonly follows Gaussian distribution. Pixel-dependent noise is more significant in sCMOS compared with EMCCD, as the variance, offset, and gain vary from pixel to pixel in sCMOS. For example, as the read noise is independent on photon numbers, the read noise levels for sCMOS and EMCCD are set at 1.5 and $0.47{\mathrm{e}}^{-}$, respectively. The fixed pattern noise for sCMOS ranges from 0.4 to $10{\mathrm{e}}^{-}$ as the photon number increases from 1 to 1500, whereas, for EMCCD, it ranges from 0.07 to $2{\mathrm{e}}^{-}$ as photon number increases from 1 to 1500. Details of the generation of the noise in accordance with the experimental noise distributions can be found in Ref. 39.

Fig. 4

Dependence of the shot noise, read noise, and fixed pattern noise on the photon number for sCMOS (a) and EMCCD (b). The shot noise and fixed pattern noise increase with the photon number for both sCMOS and EMCCD, whereas the read noise is independent of photon number. The sCMOS detector exhibits less shot noise compared with the EMCCD. However, read noise and fixed pattern noise of sCMOS are both larger than that of EMCCD.

To evaluate the effect of noise on the performance of sCMOS and EMCCD for SOFI imaging, the simulation was conducted for different photon numbers, ranging from 1 to 1500 photons per emitter. As shown in Fig. 5(a), the resolution test object features multiple spatial frequencies at different radial distances. We calculated the corresponding spatial frequency values when the correlation value equals 0.2 to evaluate the frequency transfer capability of sCMOS and EMCCD under different SNRs. The curve describing the cut-off spatial frequency versus SNR is shown in Fig. 5(b). In order to exclude the influence of different pixel sizes for sCMOS and EMCCD, we set the pixel size as 65 nm for both detectors. As can be seen, at both low and high SNR regimes, sCMOS and EMCCD have very similar performances. The differences of noise levels for sCMOS and EMCCD do not have significant influence on SOFI performance.

Fig. 5

(a) Third-order SOFI image of the resolution rest object using sCMOS camera. Dotted circles indicate the positions of different spatial frequencies. (b) Cut-off frequency vs. SNR for sCMOS and EMCCD. Shot noise, read noise, and fixed pattern noise were taken into account for evaluating the performance of SOFI imaging. The fluorescent wavelength was 625 nm, and a $100\times$ objective with numerical aperture of 1.4 was used in the simulation. 100 frames were simulated for SOFI reconstruction. The quantum efficiencies for sCMOS and EMCCD are 72% and 90%, respectively.

In Fig. 6, we compare SOFI imaging with different gain features for sCMOS and EMCCD. In EMCCD, the gain for each pixel is almost constant. In contrast, the pixels are read out independently for sCMOS. Therefore, the gain for each pixel is different. In this simulation, we set the gain for sCMOS as a random value between 9 and 11 for each pixel, and the gain for EMCCD was set as a constant of 10. From the third-order SOFI reconstructed images in Figs. 6(c) and 6(d) as well as the curve in Fig. 6(e), we can see that the gain differences between sCMOS and EMCCD do not significantly influence the imaging performances of SOFI reconstruction.

Fig. 6

Comparison of SOFI imaging with different gains for sCMOS and EMCCD. (a) and (b) Generated single frame images of sCMOS and EMCCD. (c) and (d) Third-order SOFI images of sCMOS and EMCCD. (e) Correlation value versus spatial frequency curve with different gains. In this simulation, the gain for EMCCD is set constant as 10. The gain for sCMOS is set as a random value between 9 and 11.

4.1.

Comparison of the Signal-to-Noise Ratio Between sCMOS and EMCCD

Although the SNR does not crucially affect the SOFI image quality, it is still a practical consideration to have a reasonable SNR. Unlike STORM/(F)PALM, which normally need to localize single molecule localizations precisely, SOFI does not require such high SNRs.¹⁴ We compared the SNR for sCMOS and EMCCD using Qdots 525 particles (Fig. 7). Practically, an SNR of 3 is commonly considered as the lower limit for accurate detection in SML, while for SOFI smaller SNRs may still be acceptable.¹⁴ Here, the SNR is defined as

Fig. 7

Comparison of SNRs between sCMOS and EMCCD. Upper panel: images with different frame rates using sCMOS and EMCCD on the same Qdot 525 particles. Scale bar: $1\mu \mathrm{m}$. Lower panel: the SNR values as a function of the frame rate. The EMCCD and sCMOS were cooled to $-70°\mathrm{C}$ and 0°C, respectively.

In Fig. 7, we can see that the SNR of sCMOS is consistently lower than that of EMCCD. However, for sCMOS, an acceptable SNR can also be achieved for SOFI reconstruction when the frame rate is over 600 fps using relative low illumination intensity. One should note that Qdots are much more photostable than organic dyes and fluorescent proteins, and it is quite difficult to saturate Qdots, practically. Therefore, better SNRs can be practically obtained using higher illumination intensity even for very fast frame rates. Moreover, the data transfer mechanism of sCMOS enables it to have a faster frame rate and larger field-of-view in comparison to EMCCD. Considering the fact that the SNR of sCMOS is only slightly less than EMCCD, we can conclude that sCMOS is more suitable for SOFI when both spatial and temporal resolution are equally important, as SOFI is not critical to SNR.¹⁴ Thus, using high brightness probes, e.g., Qdots and Skylan-S, a high spatiotemporal resolution of live-cell SOFI with sCMOS can be obtained.^28,29

4.2.

Comparison of Super-Resolution Optical Fluctuation Imaging Performance Between sCMOS and EMCCD

For the direct comparison of SOFI performance between sCMOS and EMCCD, the microtubule of COS 7 cell labeled with Qdots 705 on the same region was captured using both sCMOS and EMCCD detector. A total of 1200 frames of raw data (Hamamatsu ORCA Flash 4.0, $2048\times 2048\text{pixels}$, about 100 fps in sCMOS and Andor iXon DU-897, $512\times 512\text{pixels}$, about 20 fps in EMCCD) were obtained for the fourth-order balanced SOFI reconstruction (a mathematical inversion of an equation of second, third, and fourth-order SOFI results together with deconvolution) without drift correction processing. Here, we have used the same optical magnification of 150 (microscopic objective: $100\times$; relay lens: $1.5\times$) for the two detectors. As shown in Fig. 8, the SNR in EMCCD was higher than sCMOS. However, because SOFI is insensitive to noise, and because of the higher frame rate of sCMOS (less sample drift), sCMOS SOFI performs even better than the EMCCD counterpart in spatial resolution. Meanwhile, much smoother microtubule structures with improved signal to background ratio can be achieved using sCMOS SOFI with fivefold better temporal resolution and 2.64-fold larger field-of-view compared with the EMCCD SOFI result.

Fig. 8

Comparison of SOFI performance between sCMOS and EMCCD. (a) Average of 1200 sCMOS raw data. (b) Fourth-order sCMOS SOFI result. (c) Average of 1200 EMCCD raw data. (d) Fourth-order EMCCD SOFI result. (e)–(h) Magnified regions indicated by the white box in (a)–(d), respectively. (i) and (j) Cross sections (the solid lines were fitted using Gaussian fitting) indicated by the white arrows in (e) and (f) and (g) and (h), respectively. Scale bar: $10\mu \mathrm{m}$ in (a)–(d) and $1\mu \mathrm{m}$ in (e)–(h). The EMCCD ($\text{gain}=20$) and sCMOS were cooled to $-70°\mathrm{C}$ and 0°C, respectively.