Precise knowledge of the effective spatial resolution in a stimulated emission depletion (STED) microscopy experiment is essential for reliable interpretation of the imaging results. STED microscopy theoretically provides molecular resolution, but practically different factors limit its resolution. Because these factors are related to both the sample and the system, a reliable estimation of the resolution is not straightforward. Here we show a method based on the Fourier ring correlation (FRC), which estimates an absolute resolution value directly from any STED and, more in general, point-scanning microscopy image. The FRC-based resolution metric shows terrific sensitivity to the image signal-to-noise ratio, as well as to all sample and system dependent factors. We validated the method both on commercial and on custom-made microscopes. Since the FRC-based metric can be computed in real time, without any prior information of the system/sample, it can become a fundamental tool for (i) microscopy users to optimize the experimental conditions and (ii) microscopy specialists to optimize the system conditions.
© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
Fluorescence nanoscopy (or super-resolved microscopy)  has the potential to revolutionize biology and medicine, as it enables the observation of the internal nanometer-scale dynamics and structure of cells and tissues with molecular specificity and minimal invasiveness . Since the diffraction does not pose a limit anymore, all the nanoscopy techniques theoretically provide molecular spatial resolution; however, in practice, other factors limit the effective resolution of a nanoscope. Similarly to conventional microscopy, the resolution of any nanoscopy technique depends on the optical/technical characteristics of the system (objective lens, working spectral region, system’s misalignments and aberrations, photon collection efficiency, photo-detectors, etc.) and on the optical characteristics of the sample (thickness, refractive indices, scattering properties, etc.). In addition, whereas fluorophore tags contribute to the resolution of conventional microscopy mainly in terms of brightness, in nanoscopy many other photo-physical properties drive on the resolution. For example, in stimulated emission depletion (STED) microscopy [3,4], the transit absorption spectra , the photo-stability , and the lifetime  of the fluorophore are key parameters to achieve effective sub-diffraction resolution. Because of all these dependencies, the knowledge of the effective resolution of a nanoscopy experiment is uncertain [8,9], but it is essential to enable sound biological conclusions.
A purely theoretical resolution estimation based on prior information about the system, the fluorophore and the specimen, can at best provide a rough estimate. Alternatively, resolution can be evaluated employing calibration samples, consisting of nanometer-scale fluorescence structures, such as beads, line-patterns, or nanorulers . However, the resolution obtained via any calibration sample may not be in agreement with the resolution achieved for the sample-of-interest, since sample characteristics, such as scattering, refractive index, and fluorophore concentration, are different. Furthermore, also for identical fluorophores, the brightness, the photo-stability, and the photo-physical properties change depending on the sample environment. Thus, there would be a need for a robust metric for estimating the effective resolution directly from the image of the sample-of-interest (image resolution) that would be sensitive to all the above-mentioned factors. The popular full-width at half-maximum (FWHM) criterion—i.e., the FWHM of the (intensity) line profile over isolated linear or punctuated nanometer-sized structures within the image (tubulin filaments, vesicles, ribosomes, etc.)—is an attempt in this direction. However, the FWHM criterion is incomplete (it needs specific samples) and laborious and often overestimates the effective resolution (it is easy to select line cuts narrowed by noise). On the other hand, the Fourier ring correlation (FRC) analysis (Supplement 1) is complete, straightforward, and sensitive to all factors that can affect the image resolution. The FRC analysis was introduced in the field of cryo-electron microscopy [11–13], and recently it has been successfully applied to the field of single-molecule-localization (SML) microscopy [14,15]. Here we demonstrate the ability of the FRC analysis to reveal the effective resolution of any STED microscopy and, more in general, point-scanning microscopy image.
The FRC metric is based on the concept of the effective cutoff frequency. The resolution of every microscope can be described considering it as a filter that attenuates the high spatial frequencies associated with increasingly fine features. Therefore, every raw microscopy image is a low-pass filtered representation of the specimen that emphasizes coarse structures, while blurring fine ones. Due to diffraction, a conventional microscope can be considered as a short-pass filter with a fixed cutoff frequency: the sample’s frequencies beyond the diffraction limit are not transmitted to the image, and thus the resolution is fundamentally limited. A super-resolved microscope can, at least theoretically, transmit all the sample’s spatial frequencies to the image; i.e., no cutoff frequency is imposed by diffraction. In practice, the different noise sources impose an effective cutoff frequency: the highest sample’s frequency that emerges from the noise represents the effective cutoff frequency. Finally, the effective resolution can be defined as the inverse of the effective cutoff frequency. Since all factors that influence the resolution correspond to changes in the transmission of the frequencies and/or in the noise level, the FRC analysis is sensitive to all these factors.
Given two images of the same field of view but with independent noise realizations (independent images), the FRC analysis allows us to retrieve the effective cutoff frequency of the images with no prior knowledge or calibration. In a nutshell, the FRC measures the degree of correlation of the two images at different spatial frequencies [Fig. 1(a), Supplement 1, Section S1]. The resulting curve is close to unity at low spatial frequencies; for spatial frequencies higher than the effective cutoff frequency, non-correlated (independent) noise realizations dominate and the curve approaches zero. The effective cutoff frequency is the frequency at which the correlation curve drops below a given threshold [Fig. 1(a)]. In a STED microscope, the two independent images can be obtained by registering two sequential frames. However, any sample drift between the two frames would influence the FRC curve and lead to an underestimated resolution (Supplement 1, Fig. S1). We compensated for this problem by using a simple drift correction algorithm based on phase correlation (Supplement 1, Fig. S1, Supplement 1, Section S2). We also successfully explored more advanced acquisition approaches (Supplement 1, Section S3), in order to retrieve the two independent images in an increasingly parallel fashion, e.g., line-by-line, pixel-by-pixel, or pulse-by-pulse (in the case of pulsed laser implementations). In this modality, the drift correction is no longer necessary (Supplement 1, Fig. S2). Similarly, two drift-free images can be obtained sorting fluorescent photons according to their arrival times (Supplement 1, Fig. S2) or using a 50:50 fluorescent beam splitter.
First, we validated the FRC metric on microtubule (Supplement 1, Section S4) images recorded with a pulsed STED microscope (Supplement 1, Section S3, Supplement 1, Fig. S4). The FRC metric clearly reveals the diffraction-unlimited but noise-limited nature of STED microscopy [Fig. 1(b)]: for a fixed excitation beam power and for increasing STED beam power the image resolution initially improves until a value well below the diffraction-limited value of the confocal counterpart (Supplement 1, Fig. S4); the noise at high frequencies starts to dominate and the image resolution degrades with increasing . In essence, for increasing STED beam power, the stimulated emission probability increases, but the signal-to-noise ratio (SNR) decreases, due to, for example, the non-perfect “zero” intensity point of the doughnut or background signal induced by the STED beam. As expected, it is possible to recover the SNR and increase the effective image resolution by increasing the intensity of the excitation beam. In this work, we used the 1/7 fixed threshold to obtain the effective cutoff frequencies, but we achieved similar results also using the 3- and the thresholds (Supplement 1, Fig. S5).
It is very important to compare the FRC metric with the most common resolution metric used in STED microscopy, namely the FWHM of the Gaussian (or Lorentzian) fit (Supplement 1, Section S6) of the image of punctuated and isolated nanometer-sized structures [sometimes referred to as the FWHM of the point-spread function (PSF)]. Since not every sample contains punctuated structures, the method is usually applied on the image of a sparse distribution of fluorescent beads. However, fluorescent beads are usually characterized by superior brightness and photo-stability; thus the estimated resolution may not be accurate for the sample-of-interest. We compared the resolution obtained via FRC and FWHM analysis by imaging samples of 60-nm-sized Crimson fluorescent beads (Fig. 2, Supplement 1, Figs. S6, S7, S9, and S10, Supplement 1, Section S5) or 20-nm-sized ATTO 647N fluorescent beads (Supplement 1, Fig. S8). The resolution estimates obtained with fixed STED beam power and increasing excitation power reveal the higher noise sensitivity of the FRC compared to the FWHM analysis [Fig. 2(a), Supplement 1, Figs. S6, S9, and S10]; the resolution estimated via the FWHM analysis is almost constant for every SNR condition (power of the excitation beam). This can be explained considering the absence of any prior information in the FRC analysis—information that is, on the other hand, imposed when considering a Gaussian model (or Lorentzian) for fitting. The higher noise sensitivity of the FRC metrics with respect to the FWHM metrics is also demonstrated by plotting the image resolution as a function of increasing STED beam power [Fig. 2(b), Supplement 1, Fig. S7]: the improvement of resolution according to the FRC metric is slower compared to the FWHM metric, which is due to the simultaneous SNR reduction. The differences between the FWHM and FRC metrics for lower STED beam power (higher SNR conditions) can be attributed to the choice of the statistical model, Gaussian or Lorentzian, and its parameters fitting, and/or to the choice of the threshold criterion used for the FRC analysis. Last but not least, the FWHM metric is biased by the size of the structure analyzed (Supplement 1, Fig. S11). Nanorulers (or DNA origami) are emerging tools for characterizing the resolution in STED microscopy and other nanoscopy techniques . A nanoruler consists of a pair of clusters of fluorophores located at a well-known and precise distance. If the STED image is able to reveal the two clusters, the image resolution is higher than the nanoruler nominal length. We applied the FRC analysis on STED images of nanorulers with different spacing and increasing STED beam power. The FRC metric is in tune with the visual estimation, as it is only possible to discern two adjacent spots that are at a distance above the retrieved resolution (Fig. 3, Supplement 1, Figs. S12, S13, and S14). We also used nanorulers to demonstrate the sensitivity of the FRC metrics to the SNR reduction induced by photo-bleaching (Supplement 1, Fig. S15).
The high sensitivity to the SNR and the low computational cost (mainly two fast Fourier transforms) of the FRC analysis make this metric a perfect candidate for implementing real-time and auto-alignment tools for STED microscopy. The performance of any STED microscopy architecture strictly depends on its ability to match to different spatial, temporal, and spectral conditions . For example, the Gaussian excitation and the doughnut-shaped depletion intensity distributions at the focus should overlap to obtain the best SNR  and thus the best resolution [Fig. 4(a)]. Furthermore, in the case of pulsed STED microscopy implementation, the depletion pulses need to immediately follow the excitation pulses to obtain the smallest fluorescent confinement , and thus the best resolution [Fig. 4(c)]. The FRC analysis reveals both benefits and limits of the time-gated detection: the registration of the signal only after the STED beam action allows us to compensate for temporal misalignment [Fig. 4(d)] and to use sub-nanosecond pulses  (Supplement 1, Fig. S16), but the longer the delay between excitation and depletion, the lower the SNR and thus the resolution [Fig. 4(e)]. In this work we showed the ability of the FRC toolbox to retrieve the effective cutoff frequency, and thus the effective resolution of any STED microscopy image. Other methods explore the image frequency content to characterize the performance of a STED microscope [17,19,20]; however, these methods are mainly qualitative or comparative, cannot be generalized to any sample, and require a model. The FRC method does not require any prior information about the system and the sample—nor does it require a model—and provides absolute resolution estimation. We demonstrated the higher sensitivity to the SNR of the FRC method with respect to the FWHM. This sensitivity makes the FRC metric an important tool for optimizing sample preparation protocols and imaging parameters, such as the excitation and STED beam powers, the pixel dwell time (Supplement 1, Fig. S17), and the spectral detection windows. Nevertheless the FRC analysis requires a pair of independent images of the sample-of-interest; the many “parallel” acquisition modalities proposed in this work make it possible to implement a fully automatic and real-time procedure. Moreover, we have found the FRC to be sensitive to the spatial-temporal alignment conditions between the excitation and STED beams, suggesting that an auto-alignment tool based on FRC may be feasible.
It is very important to consider the possibility to apply the FRC analysis on any STED microscopy system, both custom and commercial (Supplement 1, Figs. S18 and S19), and on many other point-scanning techniques, such as confocal (Supplement 1, Figs. S4 and S9), image scanning, and RESOLFT microscopy. Future directions will be (i) the application of the FRC metric for image deconvolution/processing  and adaptive optics  and (ii) the modeling of the FRC curve to reveal other important characteristics of the image; e.g., the FRC can detect when an image has been collected under a saturate (non-linear) regime of the photo-detector (Supplement 1, Fig. S20).
In conclusion, this work proposes the FRC as a reliable, calibration-free, and user-friendly method to estimate the image resolution for STED microscopy and other diffraction-limited or diffraction-unlimited point-scanning fluorescence microscopes.
The authors thank Dr. Paolo Bianchini (Istituto Italiano di Tecnologia), Ulf Schwarz (Leica Microsystems), and Elena Tcarenkova (University of Turku) for providing Leica TCS STED CW, Leica TCS SP8, and Abberior Instruments STED images, respectively; Dr. Michele Oneto (Istituto Italiano di Tecnologia) for sample preparations; Dr. Paolo Bianchini and Prof. Colin J. R. Sheppard (Istituto Italiano di Tecnologia) for useful discussion; and Matteo Moro (University of Genoa) for helping in the experiments.
See Supplement 1 for supporting content.
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