Calibration-free estimation of field dependent aberrations for single molecule localization microscopy across large fields of view

Abstract

Image quality in single molecule localization microscopy (SMLM) depends largely on the accuracy and precision of the localizations. While under ideal imaging conditions the theoretically obtainable precision and accuracy are achieved, in practice this changes if (field dependent) aberrations are present. Currently there is no simple way to measure and incorporate these aberrations into the Point Spread Function (PSF) fitting, therefore the aberrations are often taken constant or neglected all together. Here we introduce a model-based approach to estimate the field-dependent aberration directly from single molecule data without a calibration step. This is made possible by using nodal aberration theory to incorporate the field-dependency of aberrations into our fully vectorial PSF model. This results in a limited set of aberration fit parameters that can be extracted from the raw frames without a bead calibration measurement, also in retrospect. The software implementation is computationally efficient, enabling fitting of a full 2D or 3D dataset within a few minutes. We demonstrate our method on 2D and 3D localization data of microtubuli and nuclear pore complexes over fields of view (FOV) of up to 180 μm and compare it with spline-based fitting and a deep learning based approach.

Fig. 1 |. Schematic of fitting field dependent aberrations from single molecule data.

a, The raw single molecule blinking frames are used as input. b, After segmentation into ROIs, a subset of ^~10³ ROIs across the FOV is selected for field dependent aberration estimation. c, The estimation of field dependent aberrations uses maximum likelihood estimation and consists of two optimization loops: a local update that updates the locations, photon count and background of each ROI while keeping the NAT coefficients constant, and a global update that updates the Legendre NAT coefficients while keeping the locations, photon count and background constant. By selecting different ROIs in b multiple times and repeating the estimation (c) an estimation precision for the aberrations can be calculated. d, Finally, in all ROIs emitters are localized using the vectorial PSF model (Vectorfit) with the estimated NAT coefficients as input.

Fig. 2 |. Aberration estimation from 2D and 3D experimental data of microtubili.

A sample was imaging in 3D with and in 2D without astigmatism. a, Estimated Zernike aberration surfaces from single molecules of the 2D data, compared to interpolated aberrations from bead z-stack calibration. b, Full FOV image reconstructed using Vectorfit and fitted aberrations from single molecules. c, and d, insets of b where (i) shows vector fitting and (ii) Gaussian fitting by Picasso [Picassco] e, Estimated Zernike aberration surfaces from single molecules of 3D data, compared to interpolated aberrations from bead z-stack calibration. f, Full 3D FOV image reconstructed using Vectorfit and fitted aberrations from single molecules. The 3D image is shifted 14 μm to the upper right compared to the 2D image. The false color indicates the axial distance from the nominal focus. In the 3D image, the microtubules that overlay the nucleus are visible in blue, while in the 2D image, these regions appear black, as the emitters are too much out of focus. g,h insets of f where (i) shows the Vectorfit result and (ii) with SMAP [SMAP]. i,j, xz-cross sections of the regions indicated in g,h. Scale bars: 10 μm (b,f), 500 nm (c,d,g,h,i,j)

Fig. 3 |. Aberration estimation and reconstructions for experimental 3D astigmatic data of NPCs.

a, Estimated Zernike aberration surfaces from single molecules, compared to interpolated aberrations from bead z-stack calibration. b, Standard deviation and CRLB for the estimated Zernike coefficients. The standard deviations were calculated by repeating the estimation process 30 times with different randomly selected subsets of 5,000 localizations. c(i)-(iii), Full FOV images reconstructed using fitted aberrations from single molecules, without aberrations ($A_{22}=103m\lambda$ and other Zernike modes set to zero) and FD-DeepLoc. d(i)-(iii), Zoomed views of the regions indicated by the boxes d(i)-(iii) in c. e(i)-(iii), Zoomed views of the regions indicated by the boxes e(i)-(iii) in d. f(i)-(iii), xz-cross sections of the regions indicated by the boxes f(i)-(iii) in e. Scale bars: 50 μm (c), 5 μm (d), 500 nm (e,f).