Achieving increased resolution and more pixels with Superresolution Optical Fluctuation Imaging (SOFI)

Abstract

Superresolution Optical Fluctuation Imaging (SOFI) as initially demonstrated allows for a resolution enhancement in imaging by a factor of square-root of two. Here, we demonstrate how to increase the resolution of SOFI images by re-weighting the Optical Transfer Function (OTF). Furthermore, we demonstrate how cross-cumulants can be exploited to obtain a fair approximation of the underlying Point-Spread Function. We show a two-fold increase of resolution (over the diffraction limit) of near-infrared quantum dot labeled tubulin-network of 3T3 fibroblasts.

Fig. 1

The effect of Fourier-reweighting on AC-SOFI images generated from a simulated data set. Upper panel: Conventional AC-SOFI approach for orders 2, 3 and 4 resp. Lower panel: Fourier-reweighted (using the Gaussian approximation of the PSF) AC-SOFI images featuring an increased resolution as compared to their untreated counterparts in the top panel. The AC-SOFI images shown have been generated using a Gaussian approximation for the PSF as derived from cross-cumulants as will be described later in this paper. Inset: Relative FWHM of the original image PSF versus the recovered SOFI PSF as a function of the cumulant order. Lines represent the theoretical value, circles represent the values obtained by a 2D-Gauss fit of the emitter indicated by the white arrow in the original image. Blue: Values obtained from the untreated AC-SOFI images from the upper panel generated using solely Eq. (4). Red: Values for Fourier-reweighted SOFI images following Eq. (5) and (6) by using the exact PSF (i.e. certain Bessel function of the first kind). Black: Values obtained by using a Gaussian approximation of the PSF as mentioned above in this caption. Scalebar: 1 µm.

Fig. 2

The effect of noise on Fourier reweighted SOFI images. The simulation was done with a signal to background ratio of 3 and a signal intensity of 18 counts per time bin (‘Signal’ denotes the maximum intensity of the PSF). Using these relatively weak imaging parameters it was still possible to generate SOFI images and apply Fourier reweighting to them. In the upper panel we show how the quality of the Fourier reweighted second-order AC SOFI image improves when more frames are acquired. The same holds true for the third-order SOFI image (lower panel). However, we found that the fourth-order SOFI image could not satisfactorily be generated using these imaging parameters (due to high noise content already appearing in the fourth order AC SOFI image).

Fig. 3

Schematic of various possibilities cross-cumulants can be used for. The squares represent pixels of the CCD-camera. (A) Second-order cross-correlation. By cross-correlating (indicated by arrows) two directly neighboring pixels (light blue squares) virtual, “cross-correlation” (red squares) pixels can be obtained. These pixels lie in between the physical camera pixels. (B). The same holds true for higher-order cross-cumulants. As shown on the example of the fourth order cross-cumulant. Using different combinations of the light blue pixels 16-times more virtual pixels can be generated. (C) Second-order correlation. The value for the red target pixel can be obtained by correlating various different pairs of pixels (as indicated by black and gray arrows). This way auto-correlations can be omitted completely.

Fig. 4

Comparison between distance factor corrected and uncorrected XC-SOFI images. (A) second-order XC-SOFI image not corrected for distance factor contributions. (B) Distance factor corrected second-order XC-SOFI image using relative variance minimization. Scalebar: 0.5 um.

Fig. 5

Estimation of the pixel recovery ability of XC-SOFI images. Upper panel: second order SOFI images. Lower panel: Fourth-order SOFI images. (A) resp. E): AC-SOFI image as obtained by a 2 × 2 resp. 4 × 4 binned data set. (B) resp. F): AC-SOFI image from the not-binned data set. (C) resp. G): XC-SOFI images obtained from the 2 × 2 rep. 4 × 4 binned data set using the cross-cumulant approach. (D) resp. H): Difference between B) resp. F) and C) resp. D). The simulation consists of 100,000 frames. Scalebar: 1 um.

Fig. 6

Comparison of the AC-SOFI image and XC-SOFI image. Simulation of 7 emitters (emitting at: 800 nm) placed 260 nm apart and imaged (Numerical aperture NA 1.2) on a grid which has a magnification of (160 nm/pixel). Blue line: second-order AC-SOFI image. Red line: second-order XC-SOFI image. Gray line: SOFI image as would be it obtained for a infinite spatial sampling (i.e. pixel size → 0 nm). As can be seen the XC-SOFI image carries high resolution information, which would not have been revealed in the AC-SOFI image due to the too coarse grid of the imaging system. Vertical gray lines indicate the emitter positions. Stars represent values which have been generated by cross-correlation.

Fig. 7

Tubulin network of a 3T3 fibroblast immuno-labeled with QD800 quantum dots. Top panel: A) original image taken from the average of 2000 frames of a wide-field microscope setup. White lines indicate locations where cross-section were taken for all images in the upper panel. B) XC-SOFI image featuring twice more pixels than the original image and a resolution enhancement of a factor of 2. C) Fourier-reweighted XC-SOFI image generated by using the PSF as obtained by the cross-cumulant approach. Also note that also this image has four times more pixels than the original image. Scalebar: 10 um. Lower panel: Cross-sections as taken from the upper panel. Black: interpolated original image, Blue: XC-SOFI image. Red: Fourier reweighted XC-SOFI image. a) PSF shrinks as the order increases. Lines indicating a Gauss fit and circles describe the data. The FWHM reflects the increased resolution enhancement: 1, 1.39 (~2) and 1.98 (~2) respectively b) A structure being resolved only in the Fourier-reweighted XC-SOFI image. The distance between the peaks is 220 nm (as indicated by the dashed black lines), which is approximatley 2x smaller than the Rayleigh limit for 800 nm emitting QDs and a NA 1.2 objective. c) “zoomed-out” cross-section giving a general impression of the improvement afforded by Fourier-reweighting.