Measuring image resolution in optical nanoscopy

Abstract

Resolution in optical nanoscopy (or super-resolution microscopy) depends on the localization uncertainty and density of single fluorescent labels and on the sample's spatial structure. Currently there is no integral, practical resolution measure that accounts for all factors. We introduce a measure based on Fourier ring correlation (FRC) that can be computed directly from an image. We demonstrate its validity and benefits on two-dimensional (2D) and 3D localization microscopy images of tubulin and actin filaments. Our FRC resolution method makes it possible to compare achieved resolutions in images taken with different nanoscopy methods, to optimize and rank different emitter localization and labeling strategies, to define a stopping criterion for data acquisition, to describe image anisotropy and heterogeneity, and even to estimate the average number of localizations per emitter. Our findings challenge the current focus on obtaining the best localization precision, showing instead how the best image resolution can be achieved as fast as possible.

Figure 1. The FIRE principle and trade-off between localization uncertainty and labeling density

a, All localizations are divided into two halves, the correlation of their Fourier transforms over the perimeter of circles in Fourier space of radius q is calculated, resulting in an FRC curve indicating the decay of the correlation with spatial frequency. The image resolution is the inverse of the spatial frequency for which the FRC curve drops below the threshold 1/7 ≈ 0.143, e.g. a threshold value at q = 0.04 nm⁻¹ is equivalent to 25 nm resolution b, Simulated localization microscopy image of a line pair with mean labeling density in the area occupied by the lines ρ = 2.5 · 10³μm⁻² and localization uncertainty σ = 7.6 nm (line distance 70 nm, cosine squared cross-section as defined in Supplementary Note 2). c, Theory (lines) and simulation data (circles) of constant FIRE value (20, 40, . . ., 100 nm) for line pairs as in b as a function of localization uncertainty and labeling density in the area occupied by the lines. Regions of localization uncertainty limited resolution (blue) and labeling density limited resolution (yellow) are separated by the red line ρσ² = e/6π. d, Localization uncertainty versus image resolution for different fixed total measurement times (1, 5, 10, 20 and 30 min.). Camera frame rates were varied to match the on-times of the emitter. The minima of the curves fall on the red line FIRE = 2πσ that separates the yellow region in which not enough emitters have been localized from the blue region in which the emitters have not been localized precisely enough.

Figure 2. The effect of density of localizations and data processing on resolution

a, Localization microscopy image of tubulin labeled with Alexa 647 in a HeLa cell (FIRE = 58±1 nm for the whole image). The image is obtained from 1.4 · 10⁴ frames recorded in 4 min, the localization uncertainty is σ = 9.7 nm after merging nearby localizations in subsequent image frames (Supplementary Methods, sec. 3.3) and the density of localizations is ρ = 6.0 · 10²μm⁻². b–e, Zoomed insets of two crossing lines constructed from fewer time frames showing poorer resolutions (indicated by the distance between the blue arrows). f, Resolution build-up during the entire data acquisition with FIRE = 2πσ plotted in blue, showing a transition from density limited resolution to precision limited resolution close to the end of the data acquisition. At the end of data acquisition we reach ρσ² = 0.06 on the order of the rule-of-thumb value 0.14 for the transition point (see Eq.(4)). g–i, Reconstructions by different localization algorithms showing that MLE (g, FIRE = 58±1 nm) and LS (h, FIRE = 60±1 nm) methods outperform CEN (i, FIRE = 88±2 nm). j, Localization microscopy image of actin cytoskeleton (F-actin) of a fixed HeLa cell labeled with Phalloidin coupled to Alexa 647, after correction for sample drift of ~ 70–100 nm during acquisition. The image is obtained from 5.0 · 10⁴ frames recorded in 8 min, the localization uncertainty (after merging nearby localizations in subsequent image frames) is σ = 8.0 nm and the density of localizations is ρ = 8.2 · 10³μm⁻² (ρσ² = 0.52, 2πσ = 50 nm). k–n, Zoomed insets of reconstructions before (k, m) and after (l, n) drift correction show that more detail is visible after drift correction, which is reflected by a better FIRE value for the entire image after drift correction (54±1 nm compared to 79±2 nm before drift correction).

Figure 3. Spurious correlations from two-color localization microscopy image

a, b, Overview image of a tubulin network (a) labeled with both Alexa 647 (red) and Alexa 750 (green), and inset (b) showing the quality of registration. c, The uncorrected FRC curves for the red and green channel are significantly higher than the cross-channel curve due to spurious correlations from repeated photo-activations of individual emitters, resulting in overly optimistic FIRE values for the red and green channels (25±1 nm and 34±1 nm, respectively, compared to 118±2 nm for the cross-channel). d, FRC curves corrected for spurious correlations are much closer to each other and give rise to similar FIRE values (108±1 nm for Alexa 647, 133±2 nm for Alexa 750, 121±2 nm for the cross-channel). e–g, Scaled FRC numerator curves showing a plateau for intermediate spatial frequencies, which is used to estimate the correction term and the parameter Q that determines the average number of localizations per binding site (Q = 10 for Alexa 647, Q = 18 for Alexa 750, Q = 0.3 for the cross-channel). For this correction (see Supplementary Methods, sec. 1) we used a mean and width of the distribution of localization uncertainties equal to 9.2 and 2.8 nm for Alexa 647, and 12 and 2.0 nm for Alexa 750.

Figure 4. 3D-FIRE

a, Orthogonal slices of the Fourier Plane Correlation (FPC) of a 3D localization microscopy image of a tubulin network. b–d, Cross-sections of the q_xq_z-plane, the q_yq_z-plane, and the q_xq_y-plane of the Fourier Plane Correlation, with added resolution threshold contours FPC = 1/7 (black lines). e, Representation of the 3D tubulin network, with the axial coordinate in false color. The FPC clearly shows the anisotropy of image content resulting from the line-like structure of the filaments (highest image resolution perpendicular to the filaments), as well as from the anisotropy in localization uncertainty (lowest resolution in the axial direction).