Parameter-free rendering of single-molecule localization microscopy data for parameter-free resolution estimation

Localization microscopy is a super-resolution imaging technique that relies on the spatial and temporal separation of blinking fluorescent emitters. These blinking events can be individually localized with a precision significantly smaller than the classical diffraction limit. This sub-diffraction localization precision is theoretically bounded by the number of photons emitted per molecule and by the sensor noise. These parameters can be estimated from the raw images. Alternatively, the resolution can be estimated from a rendered image of the localizations. Here, we show how the rendering of localization datasets can influence the resolution estimation based on decorrelation analysis. We demonstrate that a modified histogram rendering, termed bilinear histogram, circumvents the biases introduced by Gaussian or standard histogram rendering. We propose a parameter-free processing pipeline and show that the resolution estimation becomes a function of the localization density and the localization precision, on both simulated and state-of-the-art experimental datasets.

Supplementary figure 1b, d and f show the estimated resolution as a function of the pixel size for both rendering methods. Despite the visual similarity between the two images, the resolution estimate varies dramatically, with the standard histogram rendering producing completely underestimated resolution. The reason why we show only three datasets instead of the five shown in Fig. 1 is that no resolution estimate (i.e. curves without any local maxima) could be obtained for WGA and gp210 datasets for standard histogram rendering.  figure 2d shows how the estimated resolution changes as a function of the Kernel size used. As expected, we see that the estimated resolution is evolving almost linearly with the size of the Gaussian Kernel. As it is already discussed in greater details in [1], the Gaussian kernel is biasing the resolution estimate by filtering the Fourier space of the image, creating artificial local maxima in the decorrelation analysis. We repeated the same calculation for two other datasets (Supplementary figure 2e and f, Mito TOM22 and Tubulin AF647 respectively), and found a similar behavior. These results suggest that fixed Gaussian rendering should not be used when assessing the resolution based on decorrelation analysis.

Supplementary note 3: Simulations of Bilinear and Standard histogram rendering
To further investigate the difference between bilinear and standard histogram rendering, we performed simulations of lines with decreasing spacing (30 to 2 nm in steps of 2 nm, line width of 2 nm; see Supplementary figure 3a, pixel size of 5 nm, 10 4 localizations). To restrict the discussion to the question of rendering, we directly simulate the localization procedure and do not discuss how different SMLM analysis software might influence the results. A detailed comparison of the performance of localization software can be found in [2,3] and it is the task of the user to select the appropriate software. For each localization the x-y position is computed by randomly selecting an emitter, and adding to its ground-truth location a normally distributed offset with a set standard deviation σ gt . The result of the simulation is a list of positions that stochastically sample the underlying structure with a certain accuracy.
We performed the simulations for 3 and 6 nm localization precision, which are comparable with values reported for the experimental datasets processed in this work. Supplementary figure 3b shows the resolution evolution as a function of the number of localizations included in the rendering. For each number of localization included in the rendering, we computed the resolution as a function of the pixel size and retained the smallest estimate. We see that both rendering methods (solid curves: bilinear histogram, dashed curves, standard histogram) are forming a plateau while reaching the theoretical resolution (dashed black lines). However, standard histogram rendering requires a significantly larger number of localizations (please note the logarithmic scale) to compensate for the rounding error. We show in Supplementary figure 3c the influence of the choice of the pixel size over the resolution estimate (bilinear histogram, σ gt =3nm). We see three distinct regimes. For too large pixel size, the image is sampling limited and the resolution increases linearly with the pixel size. If the pixel size is too small, the localizations are spread and do not correlate spatially. Finally, if the number of localizations is too low, the resolution estimate will not reach its theoretical minimum.

Supplementary note 4: Bilinear rendering and resolution estimate runtime
Since the methodology we propose to estimate the resolution of SMLM datasets relies on the iterative processing of the input data, we performed runtime assessments of the rendering and the resolution estimate. All the results shown were obtained using a Windows 10 installation of Matlab (R2017b), an Intel i7-3960X CPU @ 3.3GHz and an NVIDIA GeForce GTX 480 (code available https://github.com/Ades91/ImDecorr).
Supplementary figure 4a shows the average time required to render a randomly generated localization dataset as a function of the image side length (given by the image field-of-view divided by the chosen rendering pixel size) for several numbers of localizations included in the rendering. We see that even in the extreme case of a 12'000 pixels x12'000 pixels image and 10 7 localizations, the rendering time barely Finally, the Supplementary figure 4c shows the runtime for the full processing pipeline (rendering and resolution estimation) as a function of the rendered image side length. We used the 5 shareLoc datasets as input data, using all the localizations available. We rendered a field-of-view of 12 µm x12 µm at pixel sizes ranging from 2 to 30 nm, resulting in the image side length of 12'000/30=400 pixels to 12'000/2=6'000 pixels. The light gray dashed lines corresponds to the runtime (rendering + resolution estimation) of the 5 datasets as a function of the image side length (proportional to the inverse of the pixel size). The solid line corresponds to the average runtime. Since the resolution estimation procedure requires to loop over all the pixel sizes, the total time required to get a final resolution estimate is given by the sum of runtime for each image side length. We found an average computation time of 32 ± 3 seconds (N = 5), which we believe to be representative of the minimal time required to estimate the resolution of a localization microscopy dataset using our methodology.